Question

Describe how the graph of each function is related to the graph of

$f\left(x\right)=x^2$.

$f\left(x\right)=\frac{1}{3}{x}^2$

Short answer

The graph$\mathit{f}\left(x\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}{\mathit{x}}^{\mathbf{2}}$is the graph vertically compressed from the parent function $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$.

Step-by-step solution

Step 1. State the concept.

Vertical Stretch and Vertical Compression:

$y=af\left(x\right),a>1$. Sketch the graph $f\left(x\right)$vertically by a factor of a.

$y=af\left(x\right),0<a<1$. Compress the graph $f\left(x\right)$ vertically by a factor of a.

Step 2. Graph each function. 

The graph of the functions$f\left(x\right)=x^2$ and$f\left(x\right)=\frac{1}{3}{x}^2$ is given by:

Step 3. State the interpretation of the graph.

The graph of $f\left(x\right)=a{x}^2$stretches or compress the parent graph vertically. Since $a=\frac{1}{3}$, the graph of $f\left(x\right)=\frac{1}{3}{x}^2$is the graph of $f\left(x\right)=x^2$vertically compressed. Compress the graph of $f\left(x\right)$vertically by a factor of $\frac{1}{3}$.

Therefore, the graph $f\left(x\right)=\frac{1}{3}{x}^2$ is the graph of vertically compressed from the parent function $f\left(x\right)=x^2$.