Q26. - Algebra 1

Question

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

 $f\left(x\right)=x^2$.

$f\left(x\right)=4x^2-18$

 

Step-by-Step Solution

Answer

The graph $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{18}$ is the graph of $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$ translated 18 units down and stretches vertically by a factor 4.

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 $f\left(x\right)=4x^2-18$ is given by:

Step 3. State the interpretation of the graph.

The graph of $f\left(x\right)=a{x}^2+c$ stretches or compress the parent graph $f\left(x\right)=x^2$ vertically and translated up or down.

If $\left|a\right|>1$, stretch the graph $f\left(x\right)$ vertically by a factor of $\left|a\right|$.  Since $a=4$, the graph of $f\left(x\right)=4x^2-18$ is the graph of $f\left(x\right)=x^2$ vertically stretched by a factor 5. It $c$ is positive, then translated up and if $c$ is negative, then translated down. $c=-18$, translated 18 units down.

Therefore, the graph $f\left(x\right)=4x^2-18$ is the graph of $f\left(x\right)=x^2$ translated 18 units down and stretches vertically by a factor 4.