Chapter SH, Problem 9.3.12EP

To determine

The functions in order from the most stretched vertically to the least stretched vertically graph.

Answer to Problem 9.3.12EP

  $g\left(x\right)=-x^2>h\left(x\right)=\frac{5}{3}{x}^2>f\left(x\right)=-2.5x^2$

Explanation of Solution

Given:

The given functions are:

  $g\left(x\right)=-x^2,h\left(x\right)=\frac{5}{3}{x}^2,f\left(x\right)=-2.5x^2$

Calculation:

Consider the function $g\left(x\right)=a{x}^2$ :

For this function, the graph is stretched vertically if $\left|a\right|>1$ and is compressed vertically if $0<\left|a\right|<1.$

In the function:

  $\begin{array}{l}h\left(x\right)=\frac{5}{3}{x}^2\\ a=\frac{5}{3}\\ a=1.66\end{array}$

So, the graph is stretched vertically as, $\left|a\right|>1.$

In the function:

  $\begin{array}{l}g\left(x\right)=-2.5x^2\\ a=2.5\end{array}$

So, the graph is stretched vertically as,

  $\left|a\right|>1.$

The most compressed graph is $g\left(x\right)=-x^2$ .

The next most compressed graph is $h\left(x\right)=\frac{5}{3}{x}^2$ as its value of coefficient a is less than that of $g\left(x\right)=-2.5x^2$ .

Order is:

  $g\left(x\right)=-x^2>h\left(x\right)=\frac{5}{3}{x}^2>f\left(x\right)=-2.5x^2$