Q27. - Algebra 1

Question

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

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

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

Step-by-Step Solution

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