Question

the graph of the parent quadratic function $f(x)=x^{2}$ and that of a second function of the form $g(x)=ax^{2}$ are shown. what conclusion can you make about the value of $a$ in the second function?

Explanation:

Step1: Recall quadratic stretch rules

For $g(x)=ax^2$, when $|a|>1$, the graph of $f(x)=x^2$ is vertically stretched; when $0<|a|<1$, it is vertically compressed. Also, $a>0$ opens upward, $a<0$ opens downward.

Step2: Compare given graphs

The parent function $f(x)=x^2$ opens upward. The second function opens downward (reflected over x-axis) and is narrower than the parent, so $|a|>1$.

Step3: Combine observations

Since it opens downward, $a$ is negative, and the narrow shape means $|a|>1$. So $a < -1$.

Answer:

The value of $a$ is less than -1, meaning $a < -1$.