Chapter 1.6, Problem 66E

Solution

To graph $y=x^2$ on two viewing windows and to apply a horizontal or vertical stretch or shrink to one of the graph. Also, to change the window in which the transformation is applied to get the same view as that of the unchanged one and to formulate a general rule for this observation.

In the case of both the directions, the $x$ -axis and the $y$ -axis must be scaled in the ratio $1:a$ to get the same view of the function as if it is never stretched or shrinked, where $a$ is the factor of the transformation in the vertical case or $\frac{1}{a}$ is the factor of the transformation in the horizontal case.

Given the function:

  $y=x^2$

Concept Used:

Given a function $y=f\left(x\right)$ .

Horizontal stretch or shrink:

The transformation $y=f\left(ax\right)$ shrinks $y=f\left(x\right)$ horizontally by a factor of $\frac{1}{a}$ when $a>1$ .

The transformation $y=f\left(ax\right)$ stretches $y=f\left(x\right)$ horizontally by a factor of $\frac{1}{a}$ when $0<a<1$ .

Vertical Stretch or Shrink:

The transformation $y=af\left(x\right)$ stretches $y=f\left(x\right)$ vertically by a factor of $a$ when $a>1$ .

The transformation $y=af\left(x\right)$ shrinks $y=f\left(x\right)$ horizontally by a factor of $a$ when $0<a<1$ .

Calculation:

The function $y=x^2$ is graphed in the viewing window $W_1$:

  

The function $y=x^2$ is graphed in the viewing window $W_2$:

  

A vertical stretch is applied by a factor of $2$ , $y=2x^2$ , to the graph drawn in $W_2$:

  

The final window is scaled so that one unit on the $x$ -axis is twice of that on the $y$

-axis:

  

Now, the graphs on $W_1$ and $W_2$ look alike.

Observe that the ratio of the units on the $x$ -axis to that of the $y$ -axis is $1:2$

Thus, if we performed a stretch or shrink along the vertical direction by a factor of $a$ , the $x$ -axis and the $y$ -axis must be scaled in the ratio $1:a$ to get the same view of the function as if it is never transformed.

Likewise, if we performed a horizontal stretch or shrink by a factor of $\frac{1}{a}$ , the $x$ -axis and the $y$ -axis must be scaled in the ratio $\frac{1}{a}:1$ to get the same view of the function as if it is never transformed.

Now, the ratio $\frac{1}{a}:1$ equals $1:a$ .

That is, wherever be the stretch or shrink, the $x$ -axis and the $y$ -axis must be scaled in the ratio $1:a$ to get the same view of the function as if it is never transformed.

Conclusion:

In the case of both the directions, the $x$ -axis and the $y$ -axis must be scaled in the ratio $1:a$ to get the same view of the function as if it is never stretched or shrinked, where $a$ is the factor of the transformation in the vertical case or $\frac{1}{a}$ is the factor of the transformation in the horizontal case.