Mth 163, Precalculus I

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Assignments

Module 1: Assignments 0-5
Module 2: Assignments 6-14
Module 3: Assignments 15-19
Module 4: Assignments 20-26

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Module 1 Major Quiz over Module 1 to be completed within a week of completing Assignment 6 Outline of Content: Assignments 0-5 Introductory Flow Model, Functions, Function Families A set of pendulum frequency vs. length data may be modeled by the function f = A L^-.5, where f is frequency and L is length. The number A is understood to be a constant number, and is called a parameter of the model. If we have the frequency at a known length we can substitute for f and L and determine the value of the parameter A. Substituting this value of A into the form f = A L ^-.5, we obtain a mathematical model of frequency vs. length. To the extent that we have accurate data our model will work for a pendulum of any desired length. Since f is a multiple of the -.5 power of L we say that f is a power function of L. If we observe the frequency of a pendulum at two lengths we can find the two parameters A and p for the model f = A L^p. This equation has two parameters, A and L. When we substitute the length and corresponding frequency into this form we get an equation with A and p as unknowns. If we substitute the length and frequency for our two observations we will therefore obtain two equations in the two parameters A and p, which we can then solve to obtain values of these parameters. Substituting the values we obtain for A and p into the form f = A L ^-p, we obtain a mathematical model of frequency vs. length. Since f is a multiple of the p power of L we say that f is a power function of L. If we have the form of a function, as with the form f = A L^-5 or the form f = A L^p above, then if we substitute a number of data points equal to the number of parameters in the model we will obtain a set of simultaneous equations whose number is equal to the number of parameters. We may or may not be able to obtain completely accurate solutions to these equations, but we often can solve them precisely to obtain values of the parameters. When we cannot solve the equations precisely we can almost always obtain approximate solutions. For example, the equations used to solve for the parameters of the model f = A L^p can be solved exactly for A and p. However the precise solution for p requires the use of logarithms, which may not be familiar to all students at the beginning of this course (and even most students who are famliar with logarithms will not remember the precise technique required). So most students will be unable to solve the equations exactly at this point of the course. However any student can solve the equations approximately using trial and error. and will be able to master the use of logarithms at a later point. A set of depth vs. clock time data for water flowing from a uniform cylinder through a hole at the bottom of the cylinder can be very closely modeled by a quadratic function of the form y = a t^2 + b t + c. One way to create the model is to choose three well-spaced data points to represent the data set. The t and y coordinates of these data points are substituted into the form y = a t^2 + b t + c, and the resulting equations are solved for the parameters a, b and c. These parameters are substituted back into the original form to obtain a quadratic function y(t). The resulting function y(t) will exactly fit the three selected data points, in the sense that if the t coordinate of one of the selected points is substituted for t, the resulting y value will be the y coordinate of that same point. A graph of the depth vs. clock time data set vs. the quadratic function model shows that the model stays very close to the data set. An analysis of the residuals will show whether the model deviates in a systematic way from the data set. The average magnitude of the residuals is one measure of how well the model fits the data. The pattern of the residuals is another. If the pattern of the residuals is random and the average magnitude of the residuals is small compared to the depth changes observed, the model is probably a good one. The standard measure of the closeness of the model to the data is the 'standard deviation', which can be pretty well understood to be the square root of an average of the squared residuals. A quadratic function y(t) has zeros for t values given by the quadratic formula. Depending on the value of discriminant there are no real zeros (negative discriminant), one real zero (discriminant zero) or two real zeros (discriminant positive). The graph of a quadratic function is a parabola; if the function has two distinct zeros the vertex of the parabola lies on the vertical line which is halfway between the two zeros. The location of the vertical line on which the vertex lies is easily found using the quadratic formula, whether the function has no zeros, one zero or two zeros. The vertical coordinate of the vertex is easily found by substituting the horizontal coordinate into the function. The graph points of the parabola y = a t^2 + b t + c whose horizontal coordinates lie 1 unit to the right and 1 unit to the left of the vertex have vertical coordinates a units above the vertex. Thus to get to these graph points from the vertex we think of moving over 1 unit and up a units. If a is negative, then an upward displacement of a units is actually a downward displacement. In order to find the depth at a given clock time we simply substitute the given clock time for y in the function y(t). In order to find the clock time at which the depth in the depth vs. clock time model is equal to a given value, we recall that y represents the depth. We therefore substitute the desired depth for y and solve for t. For a quadratic depth function y(t) we will use the quadratic formula to solve for the desired clock time t. Often the quadratic formula will give us two real solutions, while a depth function will only pass a given depth at one clock time. In this case we must choose the value of the clock time which is in the domain of the actual function. If the depth function is not quadratic, we will have to use another appropriate means to solve the equation. For a linear depth function the solution is very easy. Some of the methods used for other types of functions will be developed later in the course. The average rate at which a depth function y(t) changes during a time interval is equal to the change in depth divided by the duration of the time interval. This average rate is, to a first approximation, associated with the midpoint of the time interval. If the depths at the beginning and at the end of the time interval are represented by points on a graph of depth vs. clock time, then the rise of the line segment from the first to the second point represents the change in depth and the run of the segment represents the duration of the time interval so that the slope represents the average rate at which depth changes during the time interval. In general if a function y(t) represents some quantity that changes with clock time, then the average rate at which the quantity changes between two clock times is equal to the change in the quantity divided by the change in the clock time. In function notation the change in the quantity is represented by y(t2) - y(t1), the change in clock time is t2 - t1 and the average rate is ( y(t2) - y(t1) ) / (t2 - t1). This average rate is represented by the slope of the line segment between the graph points ( t1, y(t1) ) and ( t2, y(t2) ). The graph of any quadratic function can be thought of as a uniformly stretched and shifted version of the basic quadratic function y = x^2. The graph of this basic parabola can be 'fattened' or 'thinned' by stretching it in the vertical direction by the appropriate factor a , which means that each point on the parabola is moved a times as far from the x axis. If we stretch by a factor a with \| a \| > 1, each point will move further from the x axis and the resulting parabola will appear thinner. If we stretch by a factor a with \| a \| < 1, each point will move closer to the x axis and the resulting parabola will appear fatter. If a is positive, the parabola will continue to open upward, whereas if a is negative, the resulting parabola will open downward. After the parabola is stretched to the right shape, the horizontal and vertical coordinates of every point on the parabola are shifted horizontally and vertically, all by the same amount, so that the vertex ends up in the right place. If the basic quadratic function y = x^2 is stretched by factor a , then shifted horizontally through displacement h and vertically through displacement y, the resulting function will be y(t) = a ( t - h ) ^ 2 + k. This relationship is often expressed in the equivalent form y = k = a ( t - h ) ^ 2. By expanding the square we can put the function into the form y(t) = a t^2 + b t + c. By a process known as 'completing the square', we can also put y(t) = a t^2 + b t + c into the form y(t) = a ( t - h ) ^ 2 + k. We do not use this process at this stage of the course. We will use it later in relation to conic sections. The technique is also important in calculus. We understand quadratic functions and their uses better if we look at various sub-families of the family of quadratic functions. Examples of sub-families include: The set of quadratic functions with h and k both zero, which consists of all quadratic functions with vertex at the origin. The set of quadratic functions with a = 1 and h equal to some fixed value, which consists of all quadratic functions with vertex at x = h, opening upward, and which are congruent to the basic parabola y = x ^ 2. The set of quadratic functions with a = -1 and k equal to some fixed value, which consists of all quadratic functions with vertex at y = k, opening downward, and congruent to the basic parabola y = x ^ 2. Quadratic functions represent quantities whose rates change at a uniform rate. The rate at which the depth of water in the flow experiment changes is changing at a constant rate. The most typical example of a situation which can be modeled by a quadratic function is that of an object, for example an automobile, coasting down a uniform incline. The rate at which the position of the automobile changes is its velocity. The velocity of the automobile changes at a constant rate, as can be observed from the steady motion of the speedometer needle. Other function families which we will take as basic for this course include the families of linear, exponential and power functions. The basic linear function is y = x. The graph of this function is a straight line characterized by a slope of 1 and a y-intercept y = 0. If this basic function is vertically stretched by factor m,** the line remains straight and its slope becomes m and y = m * x. If the function is then vertically shifted through displacement b, every point is displaced y units in the vertical direction. Its y intercept thus becomes y = b and its formula becomes y(x) = m x + b. To graph this function we can graph the point ( 0 , b ) on the y axis and the point 1 unit to the right of this point; this second point is displaced 1 unit horizontally and m units vertically from (0 , b). A linear function will exactly fit any two given data points for which the horizontal coordinates differ. Typical situations involving linear functions include Force vs. displacement of pendulum Horizontal range of stream vs. time for flow from side of vertical uniform cylinder Income: Money earned vs. hours worked Demand: Demand for a product vs. selling price (simplified economic model) Straight-line approximation to any continuously changing quantity over a short time interval The basic exponential function is y = 2 ^ t. The graph of this function is asymptotic to the negative t axis, passes through (0,1) and grows more and more quickly, without bound, as t becomes large. Every time t increases by 1 this function doubles. The function is generalized by a compression by factor k in the horizontal direction, a vertical stretch A, and a vertical shift c. The resulting function is y(t) = A * 2 ^ ( k t ) + c. We think of first bringing every point of the y = 2 ^ t function k times closer to the y axis. If k is negative the function is also reflected about the y axis so that it becomes asymptotic to the positive t axis. We then stretch the function vertically by factor A, which changes the y intercept from (0, 1) to (0, A). We finally shift the function c units vertically, which changes the y intercept to (0, A + c) and the asymptote to the horizontal line y = c. This function also can be expressed in the form y = A e ^ (kt) + c, where A and c are the same as before and k differs. This function can be expressed in the third form y = A b ^ t + c, where A and c are the same as before and b = 2 ^ k. Typical situations involving exponential functions include Compound interest: Value of investment vs. time Unrestricted population growth: Population vs. time Temperature approach to room temperature: Temperature vs. time Radioactive decay: Amount remaining vs. time The power function family is actually a multiple family of functions characterized by basic functions of the form y = x ^ p. The power p can be any real number, either positive or negative. All basic power functions are defined at least for positive values of x, and all pass through the point (1, 1). If p is positive the function also passes through the point (0, 0). If p is negative the function has a vertical asymptote at x = 0 and approaches the x axis as an asymptote. The values of the function at x = 1/2 and x = 2, in addition to the characteristics summarized above, indicate the general shape of its graph. If p is an integer, then the function is defined for both positive and negative values of x. For even integers p the graph of the basic power function is symmetric about the y axis and has its lowest value at x = 0. A function symmetric about the y axis is also called and even function. For odd integers p the graph of the basic power function is anti-symmetric about the y axis (also described as symmetric through the origin) and has a point of inflection (where the graph changes from downward curvature to upward curvature). If p is a rational number with denominator, in lowest terms, being odd then the function is defined for both positive and negative values of x. The symmetry of the function is then determined by the whether the numerator is even or odd by the same rule as if p is and integer. If p is neither and integer nor a rational number with odd denominator, then the basic power function is not defined for negative values of x. For example a rational power with even denominator would require us to take an even root of a negative number, which we cannot do. We cannot define a real irrational power of a negative number. For each value of p there is a family of power functions y = A (x - h ) ^ p + k. As before, A is the vertical stretch applied to the basic function y = x ^ p, while h and k are the horizontal and vertical shifts. The vertical stretch A is applied first, moving every point A times further from the x axis. In particular, the point (1, 1) becomes (1, A). For positive p, the extreme point or the point of inflection will then be shifted from the origin to the point (h, k). Typical situations involving power functions include Period of pendulum vs. length (power p = .5) Frequency of pendulum vs. length (power p = -.5) Surface area vs. scale for a family of geometrically similar objects (power p = 2) Volume vs. scale for a family of geometrically similar objects (power p = 3) Strength vs. weight for geometrically and physiologically similar individuals (power p = 2/3) Illumination vs. distance from a point source (power p = -2) Illumination vs. distance from a line source (power p = -1) \
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00	qa_00	Complete Part 4 of the Initial Activities.		CD Contents	text_00				query_0

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01	qa_1	Introduction to Central Themes; Examples Central Concepts What do we need to know about mathematics? Using mathematics to solve real problems Introductory Flow Experiment (above includes link to Analyzing the Data and Understanding the Modeling Process, which students are instructed to complete as part of Assignment 1) Analyzing the Data and Understanding the Modeling Process (as instructed within the Introductory Flow Experiment)		View Precalculus material on GEN 1 CD and ponder questions posed in documentation flow experiment simulated data	text_01		Class Notes #01-02		query_1
Class Notes Topics #01: Depth vs. Clock Time Model · Depth vs. clock time possible graphs · Depth vs. clock time data · Graph of results · Calculate rates (note modify with complete definition of roc?) #02: Quadratic Model of Depth vs. Clock Time (3 points, 3 simultaneous equations) By stretching and shifting the graph of the basic parabola y = t^2, we can obtain the graph of any quadratic function y = a t^2 + b t + c. To fit a quadratic function to our depth vs. clock time data, we choose three points we believe to lie on the graph of depth vs. clock time. We substitute the coordinates of these points into the form y = a t^2 + b t + c to obtain three simultaneous linear equations in the parameters a, b and c. We solve these equations using elimination to obtain a, b and c which we then substituting to the form y = a t^2 + b t + c to obtain our quadratic model. We then evaluate the model by evaluating y each t value from our original data set, and compare these predictions of the model with the actual observe y values. The differences between predicted and observe values are called 'residuals'; we consider our model to be good if residuals are small and if there is no consistent pattern to the residuals. · y = x^2 basic, shifts and stretches yield any quadratic (asserted) · assume y = a x^2 + b x = c, select three points on curve, find three equations · solve equations, get model · find deviations and residuals
Objectives: 01.01. Know and be able to state the quadratic formula, as given in Analyzing the Data and Understanding the Modeling Process. 01.02. Solve a system of simultaneous linear equations by the process of elimination. 01.03. Given a set of y vs. t data: sketch a graph of the data sketch a smooth trendline for the data find three representative points on the trendline obtain three simultaneous linear equations for the values of a, b and c of the quadratic function model y = a t^2 + b t + c solve the equations write the model evaluate the model for appropriate t values to create a table sketch the graph of the model find deviations of the model from
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02	qa_2	Overview and Introduction: The Modeling Process applied to Flow From a Cylinder The Introductory Flow Model Exercises Summary of Modeling Process, Version 1 Summary of Modeling Process, Version 2 Summary of Modeling Process, Version 3 To Be Memorized Completion of the Introductory Flow Model Solving the system of simultaneous equations Eliminating Variables when the numbers are messy Assessing the function model The basic questions we can ask of a function model Solving the quadratic equation using the quadratic formula Exercises	Optional for feedback: Intro_Flow	View Introduction on CD #2	text_02				query_2

Objectives: 02.01. Hand-sketch a set of y vs. t data points on a y vs. t graph, sketch a reasonable smooth trendline, select three representative points, obtain three simultaneous equations for the parameters a, b and c of the quadratic model y = a t^2 + b t + c, use the model to find t for given y and y for given t, compare the model to data points and assess how well the trend of the model matches the trend of the data. More specifically: Relate the following: a set of more than three data points in a coordinate plane hand-sketch a graph and a smooth curve representing the data, selection of three representative points on the curve algebraically-determined quadratic function fitting the three selected points deviations and residuals patterns in the residuals evaluation of the quality of the model use of the model to determine the predicted value of y given the value of t use of the model to determine the value of t given the value of y the vertex of the parabolic graph of the function graph of the model constructed using stretching and shifting transformations, starting with the y = t^2 function transformed graph expressed in the notation y = A f(x - h) + k, where f(x) = x^2 Given a quadratic function in the form y = a t^2 + b t + c express the function in the standard form y = a (t - h)^2 + k.
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03	qa_3	Properties of Quadratic Functions Parabolas and Quadratic Functions The Graph of a Specific Quadratic Examples of specific quadratics Exercises 1-3 QUADEQ if practice is required dos Graphs of selected parabolas dos Most Basic Properties of Linear, Quadratic and Exponential Functions: Basic Points (defining points) of the Basic Linear Function y = x Basic Points (defining points) of the Basic Quadratic Function y = x^2 Basic Quantities (defining quantities, two points and asymptote) of the Basic Exponential Function y = 2^x			text_03		Class Notes #03		query_3
Class Notes Topics #03: Graphs of Quadratic Functions The quadratic formula tells us that the graph of y = a t^2 + b t + c will have zeros at t = [ -b +- `sqrt(b^2 - 4 a c) ] / (2 a), and know where else; that is, the graph will pass through the t axis provided these values are real numbers, and will pass through the t axis note where else. The graph will have a vertex halfway between the zeros, at t = -b / (2a); this is the coordinate of the vertex even enough there are no real zeros. The y coordinate of the vertex is easily obtained by substituting this value of t into the form y = a t^2 + b t + c. The graph points corresponding to t values which are 1 unit to the right and to the left of the vertex will lie at vertical coordinates which are a units 'up' from the vertex (if a is negative then a units up is actually down). · find zeros · find x coordinate of vertex (symmetric with zeros) and y coordinate of vertex · interpret vertex · locate points 1 unit to right and left of vertex
Objectives: 03.01: Relate {y = x, (0, 0), (1, 1), hand-sketched graph of points, hand-sketched graph of function, 'basic linear function'} 03.02: Relate {y = x^2, (-1, 1), (0, 0), (1, 1), hand-sketched graph of points, hand-sketched graph of function, 'basic quadratic function'} 03.03: Relate {y = 2^x, (-1, 1/2), (0, 1), (1, 2), hand-sketched graph of points, hand-sketched graph of function, 'basic exponential function'} 03.04: For a given quadratic function determine its zeros and its line of symmetry based on information obtained from the quadratic formula, and using this information its vertex. Relate all these quantities: 03.05: Relate { x \| a x^b + b x + c = 0 } U {quadratic formula, line of symmetry, vertex, points 1 unit right and left } 03.06: For a given quadratic function y = f(t) find the value of y corresponding to a given t, for a given value of y determine how many values of t exist such that y = f(t) and find all such values. 03.07: For a given quadratic function y = f(t) find the intersections of the graph of the function with a given horizontal or vertical line. 03.08. Construct reasonable hand-sketched graphs of the basic linear, quadratic and exponential functions y = x, y = x^2 and y = 2^x based on knowledge of the shapes of these graphs, and the coordinates of not more than three basic points.
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04	qa_4	Introduction to f(x) notation and the generalized modeling process (Class Notes #04) Introduction to function notation Examples of the f(x) notation Exercises 1-2 Introduction to the Generalized Modeling Process Detailed Summary of the Process Substitute parameters Graph the model Pose and answer questions Brief Summary of Generalized Modeling Process Exercises 3-4 Modeling Exercises Basic Point Graphs	Week 2 Quiz #1		Review of algebra in text_04		Class Notes #04		query_4
Class Notes Topics #04: Graphs of Quadratics Functions; Function Notation Graphs of Quadratics We find the graph of a specific quadratic function using the quadratic formula and what it tells us about the vertex. Function Notation We understand function notation f(x) as meaning that f(#expression#) tells us to substitute #expression# for x in the definition of f(x). Algebra Note We square the expression (a + b) using the distributive law of multiplication over addition, not by using FOIL, which should be abolished. Thus we learn what we need to know to find (a + b) ^ 3, (a + b) ^ 4, etc., and in general to multiply polynomial expressions without resorting to a mindless mnemonic which can't be generalized to anything whatsoever. · example function find zeros, vertex, y intercept · apply function notation so we can talk more generally · zeros where f(t) = 0, y intercept f(0) · single-letter vs. sensible names for functions and variables · algebra notes: squaring a binomial, avoid F-word Relate the formula for a function y = f(x), a set of x values and the corresponding set of y values.
Objectives: 04.01: Relate the formula for a function y = f(x) a set of x values the corresponding set of y values. 04.02: { x_i \| 1 <= i <= n } U {formula for f(x)} U { y_i \| y = f(x_i), 1 <= i <= n } 04.03: Relate the following: a basic linear, quadratic or exponential function the basic points of the function the basic points as transformed by a given set of vertical shifts, horizontal shifts and vertical stretches the corresponding y = A f(x-h) + c form of the transformed function the graph of the basic function the graph of its basic points the graph of the transformed function. 04.04: Relate {f(x) \| f(x) is basic linear, quadratic or exponential function} U {basic points} U {h, c, A, expression for A f(x - h) + c, transformed basic points of y = A f(x - h) + c constructed by shifting and stretching transformations basic points, graph of y = A f(x - h) + c }
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05	qa_5	Introduction to Basic Function Families Introduction The four basic functions and their families Exercise 1 Constructing Families by Specifying Parameters Exercises 2-10 Outline of the Four Basic Families The family of linear functions The family of quadratic functions The family of exponential functions The family of power functions		Basic-point graphs: Identifying Equations from Basic Points: Do quadratic functions only. Be able to quickly find the equation of a quadratic function from its basic points.	text_05		Class Notes #05		query_5
Review the Outline of Content for Assignments 1-5 (located in the Expanded Assignments grid), the Linked Outline , Class Notes, worksheets, your posted work and other resources to prepare for the Major Quiz. The Major Quiz should be completed within a week of completing Assignment 6.
Class Notes Topics #05: Introduction to Function Families Analyzing a Quadratic Function We analyze a quadratic function y = a t^2 + b t + c by determining the locations of its zeros, if any, the location of its vertex, the points 1 unit to the right and left of the vertex, and the y intercept. We can find the value of y corresponding to a given value of t by substitution; we can fine the value(s) of t corresponding to a given y using the quadratic formula. Basic Function Families The first three of the four basic functions are y = x, y = x^2 and y = 2^x. We graph these functions by first making a table for each. We see that y = x yields a straight-line or linear graph, y = x^2 yields a parabolic graph with vertex at the origin, and y = 2^x yields a graph which is asymptotic to the negative x axis and which increases more and more rapidly for increasing values of x. Stretching and Shifting the Basic Functions When a graph is stretched vertically by a given factor a, every point on the graph moves in the vertical direction until it is a time as far from the horizontal axis as it was before. If \| a \| is less than 1, every point moves closer to the horizontal axis and the graph appears to be compressed; if \| a \| is greater than 1, every point moves further from the horizontal axis and the graph appears stretched. If a < 0, then positive values of the basic function are transformed into negative values and negative values into positive, and the graph appears inverted. When the linear function y = x is vertically stretched by factor a the its slope is multiplied by a. When a graph is shifted either horizontally or vertically, every point moves the corresponding horizontal or vertical distance, and the graph simply shifts left or right, or up or down, as the case may be. When applying a series of stretches and shifts to a graph, we always apply the stretches first. Applying the shifts first would give a different result. A variety of function families can be generated by applying one or more fixed transformations, and by also applying a transformation whose parameter varies over a given range. The Number of Transformations Required to obtain the General Function for each Basic Function A linear function is characterized by its slope and y intercept. It requires only a vertical stretch of the basic y = x function to match the slope of any desired linear function, and only a vertical shift to match the y intercept. A quadratic function y = a t^2 + b t + c is characterized by the location of its vertex and by the value of a. It requires only three transformations to transform y = t^2 into a given quadratic. We first stretched the function vertically by factor a, then shift it horizontally and vertically to reposition the vertex. A general exponential function y = A * 2^(kt) + c can be obtained from y = 2^t by a vertical stretch by factor A, a horizontal stretch by factor 1/k (or the horizontal compression by factor k), and a vertical shift c. · summary of analysis of quadratic functions with example · brief review of laws of exponents · basic functions linear (y = x), quadratic (y = x^2), exponential (y = 2^x), tables -3 <= x <= 3. · sketches, characteristics of basic functions (should not doubling time of exponential) · stretching and shifting the basic functions (results yield general functions with same types as basic function) · stretch and single shift yields general linear function; stretch and two shifts required for general quadratic or exponential · general linear function defined by two points (or two parameters), general quadratic by three points (three parameters), exponential by two points and asymptote (three parameters)
Objectives: 05.01: Locate the points on a graph which represent transformed basic points of a linear, quadratic or exponential function identify the function and the transformations, and give the expression for the transformed function. 05.02: For a basic function y = f(x) sketch function families based on the form y = A f(x - h) + c. 05.03: Given A f(x-h) + c with two of the parameters A, h, c given, the third with a range of values, graph the corresponding function family.
Module 2: Assignments 6-14 Test #1 should be completed within a week of completing Assignment 14 Outline of Content, Assignments 6-13 Properties of Linear Functions We can form a good linear model of a set of data points by sketching a line which minimizes the average distance between the data points and the line. We then choose two points on the line and plug them into the y = mx + b form of the equation of a straight line and determine the parameters m and b. Given two points on a straight line we can use the slope = slope form ( y - y1) / ( x - x1) = slope, where slope = rise / run = ( y2 - y1) / ( x2 - x1). It is essential understand the geometry and the logic of this form of the equation of a straight line. Any equation in slope = slope form can be easily rearranged by solving for y to obtain the slope-intercept form. The basic points on the graph of a linear function are taken to be the y-intercept and the point 1 unit to the right and m units up from this point. For any function y = y(x), between x = x1 and x = x2 the linear function between the two corresponding graph points gives us an approximation of the original function. For most functions when x1 and x2 are sufficiently close the approximation is also close to the actual function. An equation of the form a(n+1) = a(n) + m, with an initial value a(0), generates a set of points which lie on the graph of a straight line whose slope is m and whose y intercept is a(0). Proportionality and Power Functions When we increase the scale of a solid object, we increase its height, its width and its depth by the same factor. If we think of the object is subdivided into a very large number of very tiny cubes, we see that each cube will be scaled in the same way. The volume of each tiny cube will therefore increase by the cube of the scaling factor, and the volume of the entire object will therefore increase by the cube of the scaling factor. If y is volume and x is linear dimension, then for such an object we will have y = k x^3. If we know the volume and the linear dimension for any such object, we can evaluate k and obtain the relationship between y and x at any scale. We can test a set of y vs. x data for a given y = k x^p proportionality by calculating k for different data points. If k is about the same for all points, then p is probably close to the right power. Linear dimensions (e.g., diagonals, lengths, altitudes) x, areas A and volumes V of similar real geometric objects obey the following proportionalities, which can be derived from the linear proportionalities A = k x^2 and V = k x^3: x = k A^(1/2), so that A2 / A1 = (x2 / x1)^2 x = k V^(1/3), so that V2 / V1 = (x2 / x1)^3 A = k V^(2/3), so that A2 / A1 = (V2 / V1) ^ (2/3) V = k A^(2/3)., so that V2 / V1 = (A2 / A1) ^ (2/3). These geometric proportionalities are all power functions. Solving Equations A linear equation is solved by strategically adding the same quantity to both sides of an equation and multiplying both sides by the same nonzero quantity. It is necessary to respect the distributive law of multiplication over addition, in all its manifestations. It is usually advisable to immediately multiply both sides of an equation by a common denominator for all fractional expressions, numerical or symbolic, in the equation. A quadratic equation is solved using the quadratic formula. An equation of the form x^p = c is solved by raising both sides to the 1/p power. Extraneous solutions can sometimes appear, so all solutions need to be checked by substitution into the equation. An equation of the form b ^ x = c can be solved by taking the log of both sides, obtaining x log b = log c, which has solution x = log c / log b. log c / log b is not equal to c / b (the log function doesn't cancel), nor to log (c / b) (the laws of logarithms, and the common sense of anyone who understands exponents and logarithms, contradict this common error). An equation of the form log ( x ) = b can be solved by exponentiating both sides. Sequences When the nth difference of a sequence gives us a nonzero constant, the sequence can be generated by a polynomial of degree n. The polynomial can be found by substituting n 'data points' into the form of an nth degree polynomial and solving the resulting system of simultaneous linear equations. When the ratio a(n+1) / a(n) of successive terms of a sequence is constant, the sequence is of the exponential form y = A r ^ n, where r is the common ratio. When the ratio of successive terms of the first difference of a sequence is constant, the sequence is of the exponential form y = A r ^ n + c, where r is the common ratio.
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06	qa_6	Basic Function Families The Family of Linear Functions The family of Quadratic functions The family of Exponential functions The families of Power functions Introduction to Rates and Slopes Rates and the flow model Rates and functions Rates and graphs Rates and function graphs When does 'greater' mean greater? The application of absolute values to steepness and speed. Exercises	Week 3 Quiz #1 Week 3 Quiz #2		text_06		Class Notes #06		query_6
Class Notes Topics #06: Power Functions, Rates and Slopes The Negative-Power Functions The negative-power functions y = x^-1 and y = x^-2 both have vertical asymptotes at the y axis. The y = x^-1 function is odd and approaches the positive y axis from the right and the negative y axis from the left, while the y = x^-2 function is even and approaches the positive y axis from both sides. It is important to understand that neither of these functions is defined when x = 0, because division by 0 is not defined; and that the reciprocals of numbers which approach 0 grow to unlimited magnitudes. Introduction to Rates and Slopes The average rate at which a quantity y changes with respect a quantity x is the quotient `dy / `dx, where `dy represents the change in y corresponding to the change `dx in x. This average rate is represented by the slope of the line segment connecting the corresponding two points on a graph of y vs. x. The slope is the rise / run between the points and `dy is the rise while `dx is the run. · negative-power functions, development of asymptote at zero · average rate of change of depth (with respect to clock time) is change in depth / change in clock time · calculating average rates from table of values · average rate of change represented by slope of graph
Objectives: 06.01: Explain why y = x^(-p), where p is a positive number, cannot be defined for x = 0. 06.02: Explain why the magnitude of y = x^(-1), where p is a positive number, can exceed any positive value M we might choose, no matter how large, if x is permitted to approach 0 as closely as we wish. 06.03: Explain why the graph of y = x^(-p), where p is a positive number, has the y axis as a vertical asymptote. 06.04: Explain why x^(-1/2) is not defined for negative values of x. 06.05: Construct the graphs of y = x^(-1) and y = x^(-2) using the basic points defined by x = -1, 0, 1/2, 1 and 2.
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07	qa_7	Introduction to linear functions Activity 1: Vertical position of a hanging door spring vs. weight on the spring Orientation by Prediction Making observations Organizing Data Postulating a model Selecting data points Alternative ways of finding a linear model Pose and Answer Questions			text_07	Linked Outline	Class Notes #07		query_7
Class Notes Topics #07: Rates; Modeling by a Linear Function · example of calculating a rate of change between nearby points · average rate of change changes at a constant rate · average rates on shrinking intervals yield more than twice the accuracy for twice the work · length of spring vs. number of cups of water (from lab simulation), plot indicates linear function, slope and vertical incepts interpreted
Objectives: 07.01: Relate {(t_i,y_i) \| 0 <= i <= n, y_i = f(t_i)} U {function f(t)} U {a = t_0, b = t_n, slope_i, aveRate_i, rise_i, run_i, `dt_i, `dy_i} U {a = t_0, b = t_n, a = t_0 < t_1 < … < t_n = b, partition of the interval [a, b] of the t axis}. · (t_i, y_i) is a point in the y vs. t plane, t_i < t_(i+1) · slope_i is the slope of the line segment from (t_(i-1), y_(i-1) ) to (t_i, y_i) · aveRate is the average rate of change of y with respect to t corresponding to the t subinterval [t_(i-1), t_i ) Interpret for y = depth of water in a container, t = clock time. Interpret for y = price of a stock, t = clock time. Limited vernacular example: Relate an ordered sequence of points of the y vs. t plane, the corresponding partition of an interval of the t axis, the slopes of the line segments between the points, the slope corresponding to a subinterval of the partition, the average rate of change of y with respect to t on each subinterval of the partition, the change in t and the change in y on each subinterval of the partition, and the interpretations when t is clock time and y is depth or price. 07.01: Relate the following: two points in the coordinate plane the slope of the straight line segment joining the points the equation of the straight line through the two points the general form of a point (x, y) on the line the y intercept of the line Technically: Relate { (x_1, y_1), (x_2, y_2), slope, equation of linear function, equation of line, (x, y) on line, y intercept } ... interpretations 07.01: Given a linear function and a quadratic function determine all points at which the graphs of the two functions intersect.
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08	qa_8	Basic-point graphs: Identifying Equations from Basic Points: Do linear functions only. Review for major quiz.	Week 4 Quiz #1 Week 4 Quiz #2 Week 4 Quiz #3		text_08				query_8

Objectives: Relate: a series of y vs. t data points a hand-sketched graph of y vs. t a hand-sketched linear trendline a selection of two points on the trendline two simultaneous equations for parameters of linear function through two selected points on the trendline the solution of the simultaneous equations the linear function model t value(s) corresponding to a given y value y value(s) corresponding to a given t value residual and deviation of model from each data point average deviation of model from data the trend of the residuals the slope of the linear model the vertical intercept of the linear model construction of the graph of the linear function from the basic function and its basic points Relate {data points (t_i,y_i) \| 0 <= i <= n} U {hand-sketched y vs. t graph of points, hand-sketched linear trendline, selection of two points on trendline, two simultaneous equations for parameters of linear function through two selected points} U {solution of equations, linear model, t value(s) corresponding to given y value, y value(s) corresponding to given t value} U {deviation of model from each (t_i, y_i), average deviation of model from data, trend of deviations} U {slope of linear model, vertical intercept of linear model, construction of graph of linear function from basic points} Given the symbols for the x and y coordinates of two points, write the slope of the line segment between the two points in symbolic form. Relate the following: two points in the coordinate plane the slope of the straight line segment joining the points the equation of the straight line through the two points the general form of a point (x, y) on the line the y intercept of the line points (x_1, y_1), (x_2, y_2) slope between two points the equation of the linear function whose graph passes through the points the equation of the straight line containing the points the coordinates (x, y) of an aribtrary point on the line the y intercept of the line the slope of the line segment joining (x_1, y_1) and (x_2, y_2) the slope from (x_1, y_1) to an arbitrary point (x, y) on the line the slope from (x_2, y_2) to an arbitrary point (x, y) on the line the slope = slope form of the equation of the line solution of the slope = slope equation for y the graphical representation of the points (x_1, y_1), (x_2, y_2) and (x, y) the right triangles defined by the points (x_1, y_1), (x_2, y_2) and (x, y) the slopes of the three right triangles
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09	qa_9	Symbolic Calculation of Slope Introduction Preliminary Exercise Expressing the slope when the function y = f(x) is not specified Exercises 2-3 Expressing the slope symbolically in terms of x1, x2 and f(x) Exercises 4-5 Expressing the points (x1, y1) and (x2, y2) symbolically Exercises 6-10			text_09		Class Notes #08		query_9
Class Notes Topics #08: Symbolic Slopes; Slope = Slope Equation · example: fit straight line to data, base equation on two data points, find equation, interpret for spring model · slope defined by (a, f(a)) and (b, f(b)); possible pitfalls in simplifying expression · slope = slope strategy for finding the equation of a straight line with given point and slope or two given points · deal with fractional expressions, avoid decimals · slope = slope generalized
Objectives: 09.01: Explain the meaning and implications of the statement that (x, y) lies on the line through (x_1, y_1) and (x_2, y_2) if and only if the slope from (x_1, y_1) to (x, y) is equal to the slope from (x_1, y_1) to (x_2, y_2). 09.02: Interpret and apply the statement in the preceding.
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10	qa_10	The Family of Linear Functions Introduction Basic function: y = f(x) = x Exercises 1-2 Generalized function y = f(x) = mx + b Exercises 3-4 Function families for m = const, b = const Exercise 5 Key parameters Key characteristics of graph Key points for graphing Exercises 6 Typical situations: Pendulum, Flow Range, Demand, ... Exercises 7- 11 Rate equation dy / dx = constant Sequence behavior (first difference constant) Difference equation a(n+1) = a(n) + constant Exercises 12-17 The Equation of a Line Finding the equation of a straight line from the slope and one data point Exercises 1 -3 A geometric model for the equation of a straight line through a given point with given slope (from the picture to the slope = slope formulation to the slope-intercept form) Exercises 4-8 The logic of the equation y = mx + b: Geometrical and algebraic definitions are equivalent. (the Theorem stating the equivalence between the equation and the picture, proof of the Theorem) Exercises 9-10	Week 5 Quiz #1		text_10		Class Notes #09		query_10
Class Notes Topics #09: Linear Functions · quiz find equation given points using slope=slope formulation · basic points picture briefly revisited · linear approximation of any (reasonable) function over short interval · linear function and rate equation dy/dt = constant · difference equation Explain the meaning and implications of the statement that (x, y) lies on the line through (x_1, y_1) and (x_2, y_2) if and only if the slope from (x_1, y_1) to (x, y) is equal to the slope from (x_1, y_1) to (x_2, y_2).
Objectives: 10.01: Identify quantities which are proportional to various powers of the linear dimensions of a three-dimensional geometric object, specifically to the first, second and third powers. 10.02: Given simultaneous values of y and x, and the proportionality y = k x^n, determine the value of k, use this value to model y vs. x as a power function, construct the graph of the function, find values of y given values of x, find values of x given values of y.
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11	qa_11	Introduction to Proportionality Introductory Exercise: Sand piles Building and Measuring the Sandpiles Fitting the y = a x^3 curve to the data Exercises 1-3 Pendulum Proportionalities Exercises 4-10			text_11		Class Notes #10		query_11
Class Notes Topics #10: Proportionality and Sand Piles · quiz find, interpret and apply linear force vs. displacement function given two data points · sand piles and proportionality · data set to estimate proportionality constant · comparison to DERIVE result
Objectives: 10.01. Observe and graph the dependence of the volume or mass of a sandpile to a selected linear dimension (e.g., height, diameter, circumference) of the pile. 10.02. Apply the techniques of ratio, proportionality and variation to relate a given linear dimensions of geometrically similar sandpiles to quantities related to the corresponding areas and volumes.
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12	qa_12	Power Function Proportionalities Introduction Exercises 1-3 Geometric Proportionalities and Scaling Exercises 4-7 Exercises 8-10 Area Ratios: Expanding Tiny Squares Summary: Some Simple Rules Thinking in terms of Ratios and Proportionalities Estimating Ratios and Proportionalities	Randomized Proportionality Problems Version 1 Version 2		text_12		Class Notes #11		query_12
Class Notes Topics #11: Proportionality · quiz: determining most appropriate of list of proportionalities to model data · working out randomized proportionality problem version 1 (factor by which volume changes when linear dimension changes from 7 to 8, extended to general ratio for x_1 and x_2) · {ratio of linear dimensions, ratio of areas, ratio of volumes} · ratio of pendulum periods · lifting strength vs. growing calf
Objectives: 12.01. Identify quantities which are proportional to various powers of the linear dimensions of a three-dimensional geometric object, specifically to the first, second and third powers, as well as to the -1 and -2 powers. 12.02. Given simultaneous values of y and x, and the proportionality y = k x^n, determine the value of k, use this value to model y vs. x as a power function, construct the graph of the function, find values of y given values of x, find values of x given values of y. 12.03. Given the nature of the proportionality between x and y, determine the ratio y_2 / y_1 of two y values as the appropriate power of the ratio x_2 / x_1 of the corresponding x values. 12.04. Construct the graph of the y = k x^p power function using the basic points corresponding to x = -1, 0, 1/2, 1 and 2, and using transformations construct the graph of y = A ( x - h) ^ p + c. Technically: Relate for some linear dimension x of a set of geometrically similar objects and a quantity y proportional or inversely proportional to x: the linear dimensions x_1 and x_2 of two objects and the value y_1 for that object the value y_2 corresponding to the second object the ratio of the linear dimensions the ratio of y values the ratio of x values the equation governing the proportionality the value of the proportionality constant a graph of y vs. x Relate for some linear dimension x of a set of geometrically similar objects in at least two dimensions, and a quantity y proportional or inversely proportional to the area of an object: the linear dimensions x_1 and x_2 of two objects and the value y_1 for that object the value y_2 corresponding to the second object the ratio of the linear dimensions the ratio of y values the ratio of x values the equation governing the proportionality the value of the proportionality constant a graph of y vs. x Relate for some linear dimension x of a set of geometrically similar objects in three dimensions, and a quantity y proportional or inversely proportional to the volume of an object: the linear dimensions x_1 and x_2 of two objects and the value y_1 for that object the value y_2 corresponding to the second object the ratio of the linear dimensions the ratio of y values the ratio of x values the equation governing the proportionality the value of the proportionality constant a graph of y vs. x Relate for some power p: the proportionality y = x^p x values x_1 and x_2 value y_1 corresponding to x_1 the value y_2 corresponding to the second object the ratio of the x values the ratio of the y values the value of the proportionality constant a graph of y vs. x Relate for powers p and p ': the proportionality y = k x^p x values x_1 and x_2 value y_1 corresponding to x_1 the value y_2 corresponding to the second object the ratio of the x values the ratio of the y values the value of the proportionality constant a graph of y vs. x the proportionality z = k ' y^p y values y_1 and y_2 value z_1 corresponding to y_1 the value z_2 corresponding to the second object the ratio of the y values the ratio of the z values the value of the proportionality constant a graph of z vs. y a graph of z vs. x the proportionality equation relating z and x the proportionality constant for the equation relating z and x Relate: the basic power function y = f(x) = x^p for power p the basic points corresponding to x values -1, 0, 1/2, 1, 2 a hand-sketched graph of the basic points the transformed graph y = f(x) = A (x - h)^p + c the values of A, h, c a hand-sketched graph of y = x^p a graph of the basic points of the basic function transformed to those of the function f(x) = A ( x - h)^p + c
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13	qa_13	final assignment prior to test 1 Laws of Exponents Solving Equations of the form x ^ a = b or x ^ (a/b) = c Exercises 1-4			text_13		Class Notes #12, 13		query_13
Class Notes Topics #12: Proportionality, Laws of Exponents, Sequences · quiz: surface area proportionality with volume · solving x^p = c · first and second difference of a sequence #13: Review Notes for Test · expression for slope between x = h and x = k points of f(x) = a x^2 + b x = c · linked outline · slope = slope equation for (x1, y1), (x2, y2) · (x1, y1), (x2, y2) into linear form, solve simultaneous equations to get linear equation · definition of best-fit line; approximating best-fit line as basis for linear model · basic points of linear function · average rate of change, average rate, interpretation of average rate · linear difference equation evaluated for initial condition, reveals linear pattern, reason fairly obvious from equation · `dy / `dx condition of linearity · instantaneous rate of change · rate equations · geometric and other proportionalities, relating proportionalities, approximating proportionalities, proportionalities and ratios
Objectives: 13.01: State and give illustrative examples of the laws of exponents 13.02: Apply the laws of exponents to simplification of algebraic expressions and solution of equations. 13.03: Find the set of solutions of the equation f(x) = A (x - h) ^ p + c = d for given numerical or symbolic values of A, h, c, d and p. 13.04: Find the set of points of intersection of a given function f(x) with another given function g(x) Technically: Relate the sequence { a_i \| 1<=i<=n} the 'difference sequence' { a'_i = a_(i+1) - a_i \| 1<=i<=n-1} the second-difference sequence {a''_i = a'_(i+1) - a'_i\| 1<=i<=n-2 higher-order difference sequences as specified or required the 'ratio sequency' { a_(i+1) / a_i \| 1 <= i <= n-1} the 'ratio of differences' sequence{ a'_(i+1) / a'_i \| 1 <= i <= n-1} an apparent pattern of differences and/or ratios the associated basic function the function modeling the sequence Relate { a_i \| 1<=i<=n} U { a'_i = a_(i+1) - a_i \| 1<=i<=n-1} U {a''_i = a'_(i+1) - a'_i\| 1<=i<=n-2} U ... U { a_(i+1) / a_i \| 1 <= i <= n-1} U { a'_(i+1) / a'_i \| 1 <= i <= n-1} U {clear pattern of differences and/or ratios, associated basic function} Relate {f(x) = A (x - h) ^ p + c} U {x \| f(x) = c}
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14	qa_14	Two Examples and a Picture Example 1: The Recurrence Relation a(n+1) = a(n) + 5 n, a(1) = 2. Example 2: Slope pattern of the function y = 2.5 x ^ 2 - 2.5 x + 2. Picture: Two Ways to use a Power-Function Proportionality y = x ^ p Optional practice problem set: More practice problems			text_14				query_14

Objectives: 14.01: Relate: recurrence relation for a(n) value of a(0) first m values of a(n) for specified m 14.02: Relate: quadratic function y = f(x) initial value x_0 interval `dx pattern of average slopes for successive intervals parameters of the quadratic function
Review the Outline of Content for Assignments 6-14 (located in the Expanded Assignments grid), the Linked Outline , Class Notes, worksheets, your posted work and other resources to prepare for the Major Quiz. Test 1 should be completed within a week of completing Assignment 14.
Module 3: Assignments 15 - 19 Test 2 should be completed within about a week of completing Assignment 19 Outline of Content: Assignments 15-19 Exponential Functions When a quantity Q(t), which is a function of time, increases over a given time period by amount r * Q the quantity is said to have growth rate r. To find the quantity Q(t) at the end of a period, if Q as the quantity at the beginning of the period we add r * Q to the original quantity, obtaining at the end of the period the quantity Q + r Q, or Q ( 1 + r). (1 + r) is therefore called the growth factor, because it is the factor by which we multiply the quantity to obtain its new value. If t stands for the number of periods and Q0 for the t = 0 quantity, it follows that Q(t) = Q0 ( 1 + r) ^ t. Thus if we let b stand for the growth factor 1 + r we can express Q(t) as *Q(t) = Q0 b^t. A function of this form is called an exponential function. Exponential functions arise naturally in the context of population growth, compound interest, temperature relaxation and radioactive decay, among numerous others. The most basic exponential function is taken here to be y = 2^t. When we apply a vertical stretch** by factor A we obtain y = A * 2^t. If we then apply a horizontal compression by factor k we obtain y = A * 2^(kt). When we then apply a vertical shift c we obtain the most general form *y = A 2^(kt) + c. Noting that 2^(kt) = (2^k)^t = b^t for b = 2^k, we see that the general form can be expressed as general form: y = A b^t + c. The graph of the general exponential function will approach y = c as an asymptote for either large positive t (if b < 1 or equivalently k negative) or large negative t (if b > 1 or equivalently k positive). The graph is characterized by the two basic points** (0, A+c) and (1, Ab + c) (alternatively (0, A + c) and (1, A2^k + c)). Another general form of the exponential function is Alternative General Form: y = A e^(kt) + c, where e is the limiting value of the quantity (1 + 1/n)^n as n becomes very large. e is an irrational number whose value is approximately 2.71828. e is the factor by which a principle at 100% interest, compounded continuously for 1 period, would grow in one period. It follows that if interest is compounded continuously at growth rate r the quantity at clock time t is *Q(t) = Q0 e^(rt). The defining characteristic of an exponential function is that the change in the quantity Q during a given time interval is proportional to Q itself, as in following cases: compound interest where the principle added during a period is proportional to the principle itself a population which produces new members at a rate proportional to the population itself a temperature which approaches room temperature at a rate proportional to the temperature excess relative to room temperature a radioactive substance where the number of decays in a time period is proportional to the amount of substance remaining. This behavior is analogous to that of a sequence defined by a(n) = k * a(n-1), a(0) = Q0. If the horizontal asymptote of an exponential function is known, for example in the case of an object's temperature approaching room temperature, the amount of a radioactive substance approaching 0, or the principle or population approaching 0 as we go back in time, then given two data points we can fit the general form** y = A b^t + c to the two points using a basic algebraic solution of the two resulting simultaneous equations for A and b. Using logarithms we can solve a similar set of equations for the general form y = A * 2^(kt) + c or y = A * e^(kt) + c. Logarithms and Logarithmic Functions A logarithmic function is in general defined as the inverse of an exponential function. The table of an inverse function is obtained by reversing the columns of the table of the original function, a process which yields a function if and only if the original function is either strictly increasing or strictly decreasing on its domain of definition. If the original function is denoted y = f(x) then we denote the inverse function by y = f^-1(x). Examples of inverse functions are the squaring function f(x) = x^2 (defined for nonnegative values of x) and the square root function f^-1(x) = `sqrt(x) f(x) = 3 x and f^-1(x) = 1/3 x f(x) = 10^x and f^-1(x) = log(x). The fact that these pairs of functions are inverses is plausibly demonstrated with a calculator by picking any number for x, applying the first function in a pair to x, then applying the second function of that pair to the result, showing that the second function 'undoes' the action of the first. When a function and its inverse are graphed on a set of coordinate axes on which the scales of the to axes are identical, it is seen that the reversal of coordinates results in graphs which are mirror images with respect to the line y = x, with each point on the graph of one function connected to the corresponding point on the graph of the other by an imaginary line segment which passes through the y = x line at a right angle. As mentioned above the function y = log(x) is the function which is inverse to the y = 10^x function. From this relationship and from the laws of exponents as they apply to the y = 10^x function we obtain the most basic laws of logarithms: log(10^x) = x 10^(log x) = x log(ab) = log(a) + log(b) log(a^b) = b log(a). Other laws of logarithms are also important and are easily derived from these. The base-10* logarithm is the function which is inverse to the y = 10^x function; the base-2 logarithm is inverse to the y = 2^x function; and the base-e logarithm is inverse to the y = e^x function. The base-e logarithm is generally denoted ln(x), standing for the 'natural logarithm' of x. Logarithms to other bases are defined similarly. Another useful property of logs is that log{base b}(x) = log(x) / log(b) and ln{base b}(x) = ln(x) / log(b). In order to solve an exponential equation we isolate the expression with the variable in the exponent then take the log of both sides. Given a set of exponential data for y vs. x we can linearize the data set by first subtracting the value of the asymptote from all y values, then transforming the y vs. x table into a table of log(y) vs. x. If the data is exponential and if we have subtracted correct asymptotic value, a graph of this transformed table will be approximately linear. If we then obtain the equation of the best-fit line for the graph, we can solve the equation for y. The best-fit line might e given in the form y = m x + b, in which case we need to understand that since we have graphed log(y) vs. x the equation really means log(y) = m x + b. This equation is solved for y by applying the base-10 exponential function to both sides to obtain 10^(log y) = 10^(mx + b). Applying the laws of logarithms and exponents we simplify this equation to obtain our solution. Given a set of power-function data we linearize the data set by transforming the y vs. x table into a table of log(y) vs. log(x), which if we have the correct power function yields and approximately linear graph. Since we have graphed log(y) vs. log(x) the resulting y = mx + b function really means log(y) = m log(x) + b, which is solved as in the case of the preceding paragraph by applying the base-10 exponential function to both sides. Logarithmic scales are very useful for respresenting quantities such as auditory or visual perception and other phenomena for which intensities vary over an inconceivably great range. The typical scale is the decibel scale, defined by dB = 10 log(I / Io), where Io represents some 'threshold intensity'. Given the value of Io this defining relation can yield a decibel number dB for any given intensity I, or the intensity I corresponding to any given dB level.
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15	qa_15	Introduction to Exponential Functions Introduction Compound interest: $1000 at 10% Exercises 1-4 Compound interest: principle P at rate r Exercises 5-7 Graphing principle vs. time Exercises 8-9 Doubling time at rate r Exercises 10-11 Irreducible Required Knowledge The Number e Exercises 12-14 Radioactive decay of plutonium Exercise 15 Radioactive decay: initial quantity Q0, rate r Half-life Exercises 16-17 Asymptotes Exercises 17-20 The exponential function forms y = A b^x, y = A * 2^(kx) and y = A e^(kx) xercises 21-24 Fitting an exponential function to data I: Obtaining a set of simultaneous (nonlinear) equations Exercises 25-28			text_15		Class Notes #14		query_15
Class Notes Topics #14: Exponential Functions · compound interest · increasing number of annual compounding · principle function · growth rate, growth factor · positive vs. negative growth rate, corresponding growth factors · doubling time · independence of doubling time from starting time · doubling time equation, solving by approximation
Objectives: 15.01: Relate: initial principle interest rate growth rate number of annual compoundings (including infinite number) time duration of investment doubling time principle function exponential function y = A b^t form of exponential function definition of Euler number e {initial principle, interest rate, growth rate, number of annual compoundings (including infinite), duration of investment, doubling time, principle function, exponential function, y = A b^t form of exponential function, definition of e} (t_1, y_1}, (t_2, y_2) y = A * 2^(k t) form of exponential function y = A * e^(k t) form of exponential function simultaneous equations for A and b value of y for given t value of t for given y vertically shifted generalized exponential function horizontal asymptote doubling time halflife construction of graph from basic points of basic functions using transformations slope characteristics of graph construction of graph from point and halflife or doubling time
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16		Behavior and Source of Exponential Functions The Rules of Exponents and common sense Exercise 1 The basic points of the graph of an exponential function; behavior of ratios Exercises 2-5 Difference equations and growth factors Exercises 6-10 Temperature Experiment Exponential functions and temperatures Exercises 11-12 The proportionality between rate and amount dy / dx proportional to y: Rate of change directly proportional to Quantity Exercises 13-14 Sequence analysis by ratios; general exponential functions Exercises 15-16 Basic-point graphs: Identifying Equations from Basic Points: Do exponential functions			text_16		Class Notes #15		query_16
Class Notes Topics #15: Test Questions; Behavior of Exponential Functions · review: value of f(x) for given x, x coordinate at which f(x) takes given value from graph, equation · review: average rates using function notation with meaningful variable names, interpretation · exponential functions y = 2^x, y = A * 2^x, y = A * b^x, comparative graphs · recurrence relation for exponential function · temperature vs. clock time, water in foam cup · average rate of change proportional to value of y = A b^t · temperature function: T – T_r = A b^t (T would be exponential function with nonzero horizontal asymptote)
Objectives: 16.01: Relate: vertically shifted exponential functions (i.e., generalized exponential function) horizontal asymptote hand-sketched data points hand-sketched trendline selection of two representative points on and horizontal asymptote of trendline y = A f(k x) + c form of exponential model 16.02: Relate: y = A f( k x) form of exponential function arithmetic sequence of x values sequence of corresponding y values ratio sequence for y values value of k graph slopes corresponding to members of ratio sequence ratio of graph slopes to y values difference equation with initial condition to generate y values 16.03: Relate: y = A f(k x) + c form of generalized exponential function arithmetic sequence of x values sequence of corresponding y values ratio sequence for y values ratio sequence for difference sequence of y values graph slopes corresponding to members of ratio sequence difference equation with initial condition to generate y values 16.04: Relate { (x_i, y_i), slope_i \| x_0, `dx, x_i = x_(i-1) + `dx, 1<=i<=n, y_i = A * b^x_i} U {r = y_i / y_(i-1) \| 1 <= i <= n} U { slope_i / y_i \| 1 <= i <= n} U {difference equation} 16.05: Relate {a_i \| 0 <= i <= n } U {a'_(i+1) / a'_i \| 1<=i<=n-1} U {A', b \| a'_i = A' b^i \| 1 <= i <= n} U {A, c \| A_i = A b^i + c \| 0 <= i <= n}
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17		Inverse Functions and Logarithms Introduction Exercises 1-5. The inverse of the y = x^2 function: table and graph Exercises 6-9 Exercises 10-11 The inverse of the y = 2^x function: table, graph and definition of log{base 2} (x) Exercises 12-14 Logarithms, Logarithmic Functions, Logarithmic Equations			text_17		Class Notes #16, 17		query_17
Class Notes Topics #16: Exponential Functions; Inverse Functions and Logarithms · quiz: exp fns modeling compound interest, exponential decay at continuous rate, rates of change · reversing table of function to get table of inverse function, graphing · 10^x function inverse to log(x), graphs model function from data . #17: Using Logarithms · quiz problem: construct table for base-5 log, graph · base 5 log compared to base 10 log, rule for conversion · solving exponential equations · solving doubling time equation · the number e · the natural log function · continuous interest · properties of logarithms · linearizing data
Objectives: Relate: x 2^x log[base 2](x) Relate: table of y vs. x values for function f(x) table of y vs. x values for inverse function f^-1(x) graph of points of y = f(x) vs. x graph of points of y = f^-1(x) vs. x case where f(x) = b^x case where f(x) = log[base b](x) case where f(x) = 10^x case where f(x) = log(x) case where f(x) = e^x case where f(x) = ln(x) Relate table of y vs. x values of function f(x) table of y vs. x values of inverse function f^-1(x) graph of f(x) graph of f^-1(x) for a general function f(x) and its inverse, and also where f(x) and f^-1(x) are one of the following pairs: f(x) = b^x f^-1(x) = log[base b](x) f(x) = x^2 f^-1(x) = sqrt(x) f(x) = 10^x f^-1(x) = log(x) f(x) = e^x f^-1(x) = ln(x) Relate for f(x) = log[base b](x) and a, c and p are real numbers: f(a) f(c) p f(a * c) f(a/c) f(a^p) f(a+c) p + f(a) p * f(a) Relate x 2^x log[base 2](2^x) Technically: Relate {(x_i, y_i) \| 1<=i<=n, y_i = f(x_i)} U {(y_i, x_i) \| 1 <= i <= n} U {f^-1(x) \| x_i = f^-1(y_i) } U {graph of f(x)} U {graph of f^-1(x)} U {f(x) = b^x, f^-1(x) = log[base b](x)} U {f(x) = x^2, f^-1(x) = sqrt(x)} U {f(x) = 10^x, f^-1(x) = log(x)} U {f(x) = e^x, f^-1(x) = ln(x) } Relate {f(x) = log[base b](x)} U {f(a), f(c), p, f(a * c), f(a/c), f(a^p), f(a+c), p + f(a), p * f(a) } Relate {x, 2^x, log[base 2](2^x)}
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18		Linearizing Data and Curve Fitting Introduction Exercises 1-3 Linearizing y = A t^2 with the Square Root Function Exercises 4-7 Generalizing the process of curve fitting by linearization Exercises 8-9 Exercise 10 Fitting Exponential Functions to Data Fitting an exponential function to data II: solving the equations Exercises 1-2 More Logarithmic Models Exercises 3-5			text_18		Class Notes #18, 19		query_18
Class Notes Topics #18: Logarithms; Modeling Exponential Behavior · quiz: solving exponential equations · modeling temperature vs. clock time · alternative strategies for solving exponential equations #19: Linearizing Exponential Data; Introduction to Polynomials · quiz: find exponential function given two data points and asymptote, find y for given x, find x for given y · linearizing exponential data · polynomial function as finite sum of non-negative-power functions · product of linear factors as polynomial function, graph · characterizing polynomial functions as product of linear and quadratic factors · graphing factored polynomial function
Objectives: Relate: table of y vs. x data points graph of y vs. x data points table of log(y) vs. x graph of log(y) vs. x table of y vs. log(x) graph of y vs. log(x) table of log(y) vs. log(x) graph of log(y) vs. log(x) trendline of each graph slope and vertical intercept of any linear trendline y vs. x equation corresponding to any linear trendline solution of y vs. x equation for y y = A f(k ( x - h) ) + c model of data Technically: Relate { (x_i, y_i), (log(x_i), y_i), (x_i, log(y_i)), (log(x_i, log(y_i) \| 1<=i<=n} U {graph of y vs. x, graph of y vs. log(x), graph of log(y) vs. x, graph of log(y) vs. log(x) } U {slope and vertical intercept of best-fit to any reasonably linear graph, y vs. x model of corresponding data}
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19		Analysis of a Pendulum Experiment			text_19		Class Notes #19, 20		query_19
polynomial functions ...combining functions
Class Notes Topics #19: Linearizing Exponential Data; Introduction to Polynomials · quiz: find exponential function given two data points and asymptote, find y for given x, find x for given y · linearizing exponential data · polynomial function as finite sum of non-negative-power functions · product of linear factors as polynomial function, graph · characterizing polynomial functions as product of linear and quadratic factors · graphing factored polynomial function #20: Linearizing Data · quiz: linearize given data sets · linearizing power function data
Objectives: 19.01: Relate: table of frequencies, periods, lengths of observed pendulums graph of frequency vs. period graph of frequency vs. length graph of period vs. length linearization of each graph y vs. x function for each graph x vs. y function for each graph composite function frequency(period(length)) composite function period(length(frequency)) all possible 2-function composites reconcile all composites with data Technically: Relate {frequency_i, period_i, length_i \| 1<=i<=n} U {graph of A vs. B for each combination of two of the three quantities frequency, period, length} U {linearization of each graph} U {A as a function of B for each combination} Module 4: Assignment 20-26 The Final Exam must be completed by the end of final exams

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20		Introduction to Polynomials Introduction Exercises 1-4 Graphs of Polynomial Functions Exercises 5-6 Shapes of Polynomial Graphs Introduction Polynomials of degree 3 Polynomials of degree 4 Polynomials of degree 5 The form (x-x1)(x-x1)(x-x3) The form (x-x1)(ax^2+bx+c) The form (x-x1)(x-x1)(x-x3) The form y = (x-x1)^3 The form (x-x1)(x-x2)(x-x3)(x-x4) The form (x-x1)(x-x2)(ax^2+bx+c) The form y = (a1 x^2 + b1 x + c1)(a2 x2 x^2 + b2 x + c2) The form (x-x1)(x-x2)(x-x3)(x-x4)(x-x5) The form (x-x1)(x-x2)(x-x3)(ax^2+bx+c) The form (x-x1) (a1 x^2 + b1 x + c1)(a2 x^2 + b2 x + c2) The form (x-x1)(x-x2)(x-x3)(x-x4)(x-x5) when not all zeros are distinct Exercises 1-5			text_20		Class Notes #21		query_20
Class Notes Topics #21: Graphs of Polynomials · quiz: graph a given partially factored polynomial (one quadratic factor unfactorable, another factorable) · graphing completely factored polynomials
Objectives: 21.01: Determine whether a given polynomial of degree 2 is factorable over the real numbers. 21.02: Given the factored form of any polynomial of degree 5 or less construct a graph by plotting its zeros and showing its behavior at each zero and its behaviors for large \| x \|. 21.03: Given a degree not to exceed 5, show all possible shapes of polynomial graphs of the given degree.
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21		Possible Numbers of Zeros, Approximations Possible number of zeros for a degree n polynomial Exercises 1-2 Fitting polynomials to data points Exercises 3-5 Approximating with polynomials A polynomial approximation of y = e^x Exercises 6-8 Other series Exercises 9-12			text_21		Class Notes #22		query_21
Class Notes Topics #22: Graphs of Polynomials and Power Functions · fundamental theorem of algebra · effect of degree on possible shapes of graph · graphing power functions using basic points
Objectives: 21.01. Given the degree of a polynomial, give the possible numbers and multiplicities of its zeros. 21.02. Given n points on a graph write down the set of simultaneous linear equations that could be solved to find the coefficients of a matching polynomial of degree n - 1. 21.03. Given the Taylor series for a function f(x) and the degree of the desired Taylor polynomial, write down the polynomial and use it to approximate f(x) near known points on its graph.
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22		Properties of Functions Part I Introduction Review: Graphs of negative-power functions Exercises 1-2 Stretching and shifting of graphs of negative-power functions Exercises 3-5 Generalizing to arbitrary functions y = f(x) Exercises 6-7 Basic-point Graphs for Practice			text_22				query_22

Objectives: 22.01. Based on its values or behavior at basic points corresponding to x = -1, 0, 1/2, 1, 2 graph the power function y = x^p for a given negative power of p. 22.02. Based on the basic points of the function y = x^p, use stretching and shifting transformations as appropriate to locate the corresponding points of y = f(x) = A ( x - h)^2 + c, where p is a negative power and A, h and c are given constant values.
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23		Combining Graphs of Functions Introduction More sums of two functions Dividing one function by another Reciprocal Functions Approximating the exponential function with summed graphs Seeing polynomials as summed graphs Exercises 1-8 Graphs for Practice: Graphs to combine			text_23		Class Notes #23		query_23
Class Notes Topics #23: Combining Functions: Ball down Incline and onto Floor · we work through a fairly complicated application in which the desired solution is a combination of functions involving sums, quotients and composites
Objectives: 23.01: Given graphs of functions f(x) and g(x) construct the graphs of f(x) + g(x), f(x) - g(x), f(x) / g(x) and 1 / f(x). 23.02: Given a series of power functions of the form y = A x^n, for positive values of n not exceeding 5, construct the graph of the sum of these functions.
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24		Product Functions and Composite Functions Graphs for Practice: Graphs to combine			text_24		Class Notes #24, 25		query_24
Class Notes Topics #24: Review; Practice Test #25: Combining Functions Graphically · review of construction of new functions by stretching and shifting transformations · graphical addition of two functions · graphical multiplication of two functions
Objectives: 24.01: Given graphs of functions f(x) and g(x) construct the graphs of f(x) + g(x), f(x) - g(x), f(x) / g(x), 1 / f(x), f(g(x)), g(f(x)), all for domains appropriate to the domains and ranges of f(x) and g(x).
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25					text_25				query_25

Objectives:
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26		Populations and Antibiotics Rabbits Exercises 1-3 Recurrence relation for rabbit population Exercises 4-5 Near-Exponential Behavior and Exponential Approximation Exercises 6-8 Generalized Fibonacci sequences Exercises 9-11 Antibiotics Exercises 12-16 Fluctuations with Dosage Exercises 17-23			text_26		Class Notes #26		query_26
Class Notes Topics · rabbit population models · repeated dosage to achieve maintenance level of antibiotic · difference equation models
Objectives:
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27		DERIVE strategies for curve fitting Introduction Linearizing data using DERIVE Exercises 1-2 Inverse Transformations using DERIVE Exercises 3-5			text_27		Class Notes #27, 28		query_27
Class Notes Topics #27: Review I The Modeling Process (with brief explanations) · Quadratic models based on 3 points · Linear models based on 2 points · Linear models based on regression line · Power-function models · Polynomial Models · Exponential Models based on 2 points and asymptote · Linearizing data · Exponential Model by linearizing exponential data · Power-function Model by linearizing power-function data · Modeling sequences by functions · Interpretation and description of graphs · Interpretation of slopes Equation Solving (problems to be solved; solutions given in #28) · Solving Linear Equations · Solving Quadratic Equations · Solving Equations with Denominators · Solving Equation x^p = c · Solving Systems of Simultaneous Equations · Solving Exponential Equations · Solving Logarithmic Equations Properties of Functions (problems to be solved; solutions given in #28) · The four basic functions · Inverse functions · Stretching and shifting transformations · Power functions · Polynomial Functions · Composite Functions · Constant Multiples, Sums and Differences of Functions · Products of Functions
Objectives:
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28		The Illumination Experiment Introduction Acquiring data Analyzing data A proportionality model Using the Model A line source			text_28				query_28
Review the Outline of Content for eac module (located in the Expanded Assignments grid), the Linked Outline , Class Notes, worksheets, your posted work and other resources to prepare for the Final Exam. The Final Exam must be completed by the last day of the exam period for the present term.
Objectives:

Major Quiz over Module 1 to be completed within a week of completing Assignment 6

Outline of Content: Assignments 0-5

Central Concepts What do we need to know about mathematics? Using mathematics to solve real problems

Solving the system of simultaneous equations

Eliminating Variables when the numbers are messy

Assessing the function model

The basic questions we can ask of a function model

Solving the quadratic equation using the quadratic formula

Exercises

Parabolas and Quadratic Functions

The Graph of a Specific Quadratic

Examples of specific quadratics

Exercises 1-3

QUADEQ if practice is required dos

Graphs of selected parabolas dos

Introduction to f(x) notation and the generalized modeling process (Class Notes #04)

Introduction to function notation

Examples of the f(x) notation

Exercises 1-2

Introduction to the Generalized Modeling Process

Detailed Summary of the Process

Substitute parameters

Graph the model

Pose and answer questions

Brief Summary of Generalized Modeling Process

Exercises 3-4

Modeling Exercises

Introduction

The four basic functions and their families

Exercise 1

Constructing Families by Specifying Parameters

Exercises 2-10

Outline of the Four Basic Families

The family of linear functions

The family of quadratic functions

The family of exponential functions

The family of power functions

Test #1 should be completed within a week of completing Assignment 14

Outline of Content, Assignments 6-13

The Family of Linear Functions

The family of Quadratic functions

The family of Exponential functions

The families of Power functions

Rates and the flow model

Rates and functions

Rates and graphs

Rates and function graphs

When does 'greater' mean greater? The application of absolute values to steepness and speed.

Exercises

Review for major quiz.

Introduction

Basic function: y = f(x) = x Exercises 1-2

Generalized function y = f(x) = mx + b Exercises 3-4

Function families for m = const, b = const

Exercise 5

Key parameters

Key characteristics of graph

Key points for graphing Exercises 6

Typical situations: Pendulum, Flow Range, Demand, ...

Exercises 7- 11

Rate equation dy / dx = constant

Sequence behavior (first difference constant)

Difference equation a(n+1) = a(n) + constant

Exercises 12-17

Finding the equation of a straight line from the slope and one data point

Exercises 1 -3

A geometric model for the equation of a straight line through a given point with given slope (from the picture to the slope = slope formulation to the slope-intercept form)

Exercises 4-8

The logic of the equation y = mx + b: Geometrical and algebraic definitions are equivalent. (the Theorem stating the equivalence between the equation and the picture, proof of the Theorem)

Exercises 9-10

Introductory Exercise: Sand piles

Building and Measuring the Sandpiles

Fitting the y = a x^3 curve to the data

Exercises 1-3

Pendulum Proportionalities

Exercises 4-10

Laws of Exponents

Solving Equations of the form x ^ a = b or x ^ (a/b) = c

Exercises 1-4

Example 1: The Recurrence Relation a(n+1) = a(n) + 5 n, a(1) = 2.

Central Concepts
What do we need to know about mathematics?
Using mathematics to solve real problems