We first predict how the depth of water in a uniform cylinder, with water escaping through a hole in the bottom of the cylinder, will change with respect clock time. We then observe depth vs. clock time for a real flow. Making a table of depth vs. clock time, we note that the rate of depth change varies progressively with clock time and changing depth. We graph depth vs. clock time and note how this varying rate determines the shape of the graph. We also calculate the average rate at which depth changes between each pair of successive data points.
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.
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 nowhere 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).
#04We find the graph of a specific quadratic function using the quadratic formula and what it tells us about the vertex.
Function NotationWe understand function notation f(x) as meaning that f(#expression#) tells us to substitute #expression# for x in the definition of f(x).
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.
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.
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.
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.
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.
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.
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.