Chapter 8.2, Problem 34E

To determine

if the given statement is correct.

Answer to Problem 34E

No

Explanation of Solution

Given:

The statement: “In equation $y=a{x}^2+c$ , the vertex changes when the value of a changes.”

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).

Vertical translation:

The graph of $g\left(x\right)=f\left(x\right)+k$ is a vertical translation of the graph of f.

The vertical shift is k units up if $k>0$ and k units down if $k<0$ .

For the equation $y=a{x}^2+c$ , the vertex is at $\left(0,c\right)$ .

Now, the change in value of a will result in the vertical transformation of the graph, and thus will only make the graph wider or narrower and can also cause its reflection about the x-axis if $a<0$ . But will for sure not change its vertex.

The vertex of the graph of $y=a{x}^2+c$ will change only when the value of c changes.

Conclusion:

The given statement is incorrect.