# Transforming functions by rescaling axes.

Introduction

High school algebra and precalculus courses introduce students to the notion of a parent function and a class of functions that can be derived from it through translation, compression, dilation, and reflection. The latter functions are referred to in this paper as generalised parent functions. The parent function can be one of a broad variety of functions, e.g., trigonometric, exponential, logarithmic, polynomial, absolute value, etc.

Students at the top-level high schools in my area are required to be able to plot a given generalised parent function by hand, based on their knowledge of the parent function plot and how the generalised parent function's parameters affect it. The requirement reflects the view that hand-plotting provides more insight because students must understand the transformations implied by the generalised parent function's parameters.

One approach students are taught to plot the generalised parent function by hand is to:

* plot the parent function for a few v-axis values to see its shape;

* transform the chosen parent function points into corresponding points that lie on the generalised parent function curve, by substituting the v-axis values into the generalised parent function;

* plot the transformed points to get an idea of the generalised parent function's shape;

* sketch the generalised parent function using the transformed points as a guide.

This approach is easily implemented in Excel, but does not provide much insight about how the generalised parent function's parameters change the parent function's shape. A more insightful approach the students are taught is to:

* plot the parent function for a few v-axis values to see its shape;

* sketch the parent function;

* repeatedly redraw the sketch to incorporate, in turn, any x-axis translation, compression or dilation, reflection, etc., implied by the generalised parent function's parameters;

* repeatedly redraw the sketch to incorporate, in turn, any y-axis compression or dilation, translation, reflection, etc., implied by the generalised parent function's parameters;

* recognise the final sketch as the generalised parent function.

This approach is easily implemented in Excel by plotting on the same chart a series of functions obtained by incorporating each of the generalised parent function's parameters one at a time in the proper order. It provides insight about how the generalised parent function's parameters change the parent function's shape.

Students sometimes have difficulty with these approaches, because the precise shape of the generalised parent function may not be obvious to them from the few transformed points or they may not be sure of the impact of the generalised parent function's parameters. There is an alternative handplot approach to seeing what the generalised parent function looks like that is simple, foolproof, does not even require plotting the generalised parent function, provides much of the same insight, and that the student can use to identify the generalised parent function's shape and verify that he has correctly performed the above procedures. The alternative approach takes advantage of the fact that a rescaling of the parent function plot's axes changes it into the generalised parent function plot. Two benefits of the alternative approach are that it makes obvious to students that generalised parent functions are equivalent to parent functions and illustrates the impact of the generalised parent function's parameters. The equivalence is clear because rescaling the axes results in the same shape for both the generalised parent function and the parent function.

The alternative approach

Consider a parent function, f(x), plotted as:

y = f(x) (1)

The generalised parent function is assumed to take the form:

(y - D) = Af(B(x - C)) (2)

The alternative approach to plotting the generalised parent function is to:

* plot the parent function;

* rescale the x- and y-axes using the following procedure.

1. Divide the x-axis values by B.

2. Add C to the new x-axis values.

3. Multiply the y-axis values by A.

4. Add D to the new y-axis values.

The result is a plot of the generalised parent function. The equivalence of the generalised parent function and the parent function is clear because both will have the same shape.

An educational way to accomplish the alternative procedure is to begin with a plot of the parent function on the x- and y-axes, with the original x- axis values at the bottom of the chart and the original y-axis values at the left side of the chart. If this is done in Excel, print a full-page copy of the plot. Then:

* Apply step 1 to a suitable set of the original x-axis values and add the new x-axis values as a second row of x-axis values below the original x-axis values. Refer to the new x-axis values as [x.sub.1]-axis values.

* Apply step 2 to the [x.sub.1]-axis values and add them as a third row of x- axis values below the [x.sub.1]-axis values. Refer to the new x-axis values as [x.sub.2]- axis values.

* Apply step 3 to a suitable set of the original y-axis values and add the new y-axis values as a second column of y-axis values to the left of the original y-axis values. Refer to the new y-axis values as [y.sub.1]-axis values.

* Apply step 4 to the y 1-axis values and add them as a third column of y-axis values to the left of the [y.sub.1]-axis values. Refer to the new y-axis values as [y.sub.2]-axis values.

The impact of one of the generalised parent function's parameters can be seen at each step.

Example 1

Consider the following parent function and generalised parent function.

y = cos (x) (3)

y = 4 cos (2[pi] (x - 3)) + 5 (4)

Figure 1 is an Excel plot of these functions. Note that, except for axis scaling, the plot of the generalised parent function from x = 3 to x = 4 is the same shape as the plot of the parent function from x = 0 to x = 2[pi]. This implies that suitably rescaling the axes will result in the rescaled generalised parent function plot looking exactly like the parent function plot for the x-axis range from 3 to 4. The rescaling procedure accomplishes this.

The rescaling procedure is illustrated below. Begin with the following Excel parent function plot.

Rescaling the x-axis

Choose a suitable set of original x-axis values for the parent function. For this example, consider the following original x-axis values.

0 [pi]/2 = 1.571 [pi] = 3.142 3[pi]/2 = 4.712 2[pi] = 6.283

Apply step 1 to get the second row [x.sub.1]-axis values:

0/2[pi] = 0 ([pi]/2)/2[pi] = 1/4 [pi]/2[pi] = 1/2 (3[pi]/2)/2[pi] = 3/4 2[pi]/2/2[pi] = 1

Apply step 2 to get the third row [x.sub.2]-axis values. These are the final transformed x-axis values.

0 + 0 = 3 1/4 + 3 = 3 1/2 + 3 = 3 3/4 +3 = 3 1 + 3 = 4

Rescaling the y-axis

Choose a suitable set of original y-axis values for the parent function. For this example, consider the following original y-axis values.

-1 -/2 0 1/2 1

Apply step 3 to get the second column [y.sub.1]-axis values.

-1(4) = -4 -1/2 (4) = -2 0(4) = 0 1/2 (4) = 2 1(4) = 4

Apply step 4 to get the third column [y.sub.2]-axis values. These are the final transformed y-axis values.

-4 + 5 = 1 -2 + 5 = 3 0 + 5 = 5 2 + 5 = 7 4 + 5 = 9

The rescaled plot, using the final transformed axis values is shown in Figure 3. Note that only the axis values have been changed, not the plot. The generalised parent function plot now looks exactly like the parent function plot. However, the rescaled axes provide the proper x- and y-axis values.

Why the axis rescaling procedure works

The parameters A, B, C, D in Equation (2) have the following effects.

* Suppose that C = 0. Then B can be viewed as compressing toward (B > 1) or expanding away from (B < 1) the plot with respect to x = 0 by a factor of B. For example, relative to the original origin, the value of the function originally achieved at [x.sub.0] is now achieved where B[x.sub.1] = [x.sub.0], i.e., at [x.sub.1] = {[x.sub.0]/B} . Hence the new x values are smaller if B > 1 and larger if B < 1. An equivalent view is that B compresses or expands the x-axis, i.e., rescales the x-axis by dividing the axis values by B. If C [not equal to] 0, B can be viewed as compressing or expanding the plot toward or away from x = C.

* C moves the x = 0 point to x = C, hence translates the origin and the plot to the right C units. For example, the value of the function originally achieved at [x.sub.1] now is achieved when [x.sub.2] - C = [x.sub.1], i.e., at [x.sub.2] = [x.sub.1] + C. This is equivalent to rescaling the x-axis by adding C to the x-axis values.

* Suppose that D = 0. Then A can be viewed as vertically compressing toward (A < 1) or expanding away from (A > 1) the plot with respect to y = 0 by a factor of A. For example, an original function value of [y.sub.0] becomes [y.sub.1] = A[y.sub.0]. This is equivalent to rescaling the y-axis by multiplying the y-axis values by A.

* D translates the origin D units upward, hence moves the plot upward D units. For example, an original value of [y.sub.1] becomes [y.sub.2] = [y.sub.1] + D. This is equivalent to rescaling the y-axis by adding D to the y-axis values.

Rescaling the x-axis

Denote the value of the parent function at [x.sub.0] by [y.sub.0] = f([x.sub.0]). The new x-axis value, [x.sub.2], corresponding to [x.sub.0] must establish a correspondence between the generalised parent function's value at [x.sub.2] and the parent function's value at [x.sub.0]. Ignoring the parameters that transform the y-axis, the two values must be equal. This is accomplished by equating the arguments of Equation (5), which then provides the x-axis transformation equation.

f (B ([x.sub.2] - C)) = f ([x.sub.0]) (5)

B ([x.sub.2] - C) = [x.sub.0] (6)

[x.sub.2] = ([x.sub.0]/B) + C (7)

Rescaling the y-axis

Consider a parent function value of [y.sub.0]. The generalization transforms this to [y.sub.2] = A[y.sub.0] + D. Thus, the transformed y-axis value corresponding to [y.sub.0] must be:

[y.sub.2] = A[y.sub.0] + D (8)

Substituting the transformed point in the generalised parent function restores the original point in the function.

The transformed point is ([x.sub.2],[y.sub.2]) = ((([x.sub.0]/B)+C)),(A[y.sub.0] + D)) and the original point is ([x.sub.0], [y.sub.0]).

((A[y.sub.0] + D) - D) = Af (B((([x.sub.0]/B)+C)-C)) (9)

A[y.sub.0] = Af (B([x.sub.0]/B)) (10)

[y.sub.0] = f([x.sub.0]) (11)

Subtleties

Suppose B < 0. Then transforming ([x.sub.0] - C) to

([x.sub.0] - C) / B

reflects the function across the vertical line x = C in addition to rescaling the x-axis. However, applying steps 1 and 2 does not move the plot. Steps 1 and 2 accomplish the equivalent by reflecting the x-axis across the vertical line x = C and rescaling it. x-axis values will now decrease to the right and increase to the left. To see how the generalised parent function looks when plotted with a normal x-axis scale:

* Hold the plot paper in front of a light (so the light is behind the paper).

* Rotate the paper around the y-axis by 180 degrees, i.e., turn it over from right to left as if you were turning a page in a book. The plot is now on the back of the paper. Viewed through the paper, the x-axis values now increase to the right and decrease to the left and the generalised parent function has its proper shape.

This subtlety is illustrated by the following example.

Example 2

Consider the following parent function and generalised parent function. The only important change from the previous example is that the generalised parent function has been reflected about the line x = 3.

y = sin (x) (12)

y = 4 sin (-2[pi] (x - 3)) + 5 (13)

These functions are plotted in Figure 4.

The rescaled generalised parent function is plotted in Figure 5. Note the reflection of the x-axis around the vertical line x = 3. x-axis values decrease to the right. Rotate the paper around the y-axis by 180 degrees, i.e., turn it over from right to left as if you were turning a page in a book. The plot is now on the back side of the paper. Viewed through the paper, the x-axis values now increase from 2 to 3 to the right and the plot is exactly the shape of the negative of the sine function.

Suppose A < 0. Then transforming ([y.sub.0] - D) to A ([y.sub.0] - D) reflects the function across the horizontal line y = D, in addition to rescaling the y-axis. However, applying steps 3 and 4 does not move the plot. Steps 3 and 4 accomplish the equivalent by reflecting the y-axis across the horizontal line y = D and rescaling it. y-axis values now increase in the downward direction and decrease in the upward direction. To see how the generalised parent function looks when plotted normally:

* Hold the plot paper in front of a light (so the light is behind the paper).

* Rotate the paper around the x-axis by 180 degrees, i.e., turn it over from top to bottom as if it is in a flip chart and you are turning to the previous page. The plot is now on the back side of the paper. Viewed through the paper, the y-axis values now increase in the upward direction and decrease in the downward direction.

* Viewing the generalised parent function plot through the paper (from the back) shows how it would look if the y-axis had been re-plotted by reversing the y-axis scale so that the y-axis values increased in the upward direction.

This subtlety is illustrated by the following example.

Example 3

Consider the following parent function and generalised parent function.

y = sin (x) (14)

y = -4 sin (2[pi] (x - 3)) + 5 (15)

These functions are plotted in Figure 6.

The rescaled generalised parent function is plotted in Figure 7. Note the reflection of the y-axis around the horizontal line y = 5. y-axis values decrease in the upward direction. Rotate the paper around the x-axis by 180 degrees, i.e., turn it over from top to bottom as if it is in a flip chart and you are turning to the previous page. The plot is now on the back side of the paper. Viewed through the paper, the y-axis values now increase from 0 to 10 in the upward direction.

Conclusion

Students are often asked to plot a generalised parent function from their knowledge of a parent function. One approach is to sketch the parent function, choose a few points on the parent function curve, transform and plot these points, and use the transformed points as a guide to sketching the generalised parent function. Another approach is to modify the parent function sketch to reflect, in turn, the impact of each of the generalised parent function's parameters. Students sometimes have difficulty with these approaches, because the precise shape of the generalised parent function may not be obvious to them from the few transformed points or they may not be clear about the impact of the generalised parent function's parameters. This paper presented an alternative approach to seeing what the generalised parent function looks like that is simple, foolproof, and does not even require plotting the generalised parent function. The alternative approach takes advantage of the fact that a rescaling of the parent function's axes changes it into the generalised parent function. Two benefits of the alternative approach are that it makes obvious to students that generalised parent functions are equivalent to parent functions and illustrates the impact of each of the generalised parent function's parameters.

Robert Ferguson

Coral Gables, Florida, USA

bobferg13@comcast.net

Caption: Figure 1. cos (x) and y = 4 cos (2[pi] (x - 3)) + 5.

Caption: Figure 2. cos (x).

Caption: Figure 3. 4 cos (2[pi] (x - 3)) + 5.

Caption: Figure 4. sin (x) and 4 sin (-2[pi] (x - 3)) + 5.

Caption: Figure 5. 4 sin (-2[pi] (x - 3)) + 5.

Caption: Figure 6. sin (x) and -4 sin (2[pi] (x - 3)) + 5.

Caption: Figure 7
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