Chapter 8.1, Problem 28E

To determine

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

Answer to Problem 28E

Always true

Explanation of Solution

Given information:

The statement: “The graph of $f\left(x\right)=a{x}^2$ is wider than the graph of $g\left(x\right)=x^2$ when $0<\left|a\right|<1$ ”

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 $f\left(x\right)=a{x}^2$ is the vertical transformation of the graph of $g\left(x\right)=x^2$ .

By the above definition, it can be said that when $0<\left|a\right|<1$ , the graph of $f\left(x\right)=a{x}^2$ is vertical shrink of the graph of $g\left(x\right)=x^2$ and thus will always be wider.

So, it can be said that the given statement is always true.