Precalculus I

Framework and Main Concepts

This document provides an overview of the framework of Precalculus I. You will work through the details in the text, in class notes, and in communication with the instructor.

If you have already had Precalculus I, this document should provide a brief review of the main topics.

If you are just starting Precalculus I, you should read over this document, though you might not understand much of it.

You should then read over the document once a week.

You will see your progress in your ability to understand what is said here, and in your knowledge of the details behind these statements.

If you are reviewing for a test, this document is a good place to start.

Topics between double asterisks ** ... ** are to be regarded as useful, maybe interesting, and instructive but not required. That is, it won't be on the test but if you understand it, it might come in handy some day.

Topics included on the major quiz are indicated by blue text within the outline.

Introductory Flow Model, Functions, Function Families

Properties of Linear Functions

Proportionality and Power Functions

Sequences

Solving Equations

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)

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.

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, A*b + c) (alternatively (0, A + c) and (1, A*2^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(a*b) = 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.