Chapter 8.1, Problem 29E

To determine

To verify: if the given statement is always, sometimes or never true.

Answer to Problem 29E

Never true

Explanation of Solution

Given information:

The statement: “The graph of $f\left(x\right)=a{x}^2$ is wider than the graph of $g\left(x\right)=d{x}^2$ when $\left|a\right|>\left|d\right|$ ”

Concept used:

Vertical transformation:

Relative to the graph of $f\left(x\right)$ , the transformation of graph of $g\left(x\right)=cf\left(x\right)$ is a vertical shrink when $0<c<1$ and is a vertical stretch when $c>1$ (where each y value is multiplied by c ).

Calculation:

It can be observed that the graph of $f\left(x\right)=a{x}^2$ and the graph of $g\left(x\right)=d{x}^2$ are both the vertical transformation of the graph of $h\left(x\right)=x^2$ .

Consider,

  $\begin{array}{l}f\left(x\right)=\frac{a}{d}\left(d{x}^2\right)\\ f\left(x\right)=\frac{a}{d}g\left(x\right)\end{array}$

Now whenever $\left|a\right|>\left|d\right|\Rightarrow \frac{\left|a\right|}{\left|d\right|}>1$ , the graph of $f\left(x\right)=a{x}^2$ is vertical stretch of the graph of $g\left(x\right)=d{x}^2$ and thus will always be narrower.

So, it can be said that the given statement is never true.