Question

Describe how the graph of each function is related to the graph$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$.

$\mathit{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}$

Short answer

The graph $\mathit{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}$is the graph role="math" localid="1647861887784" $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$ vertically compressed by a factor $\frac{\mathbf{1}}{\mathbf{2}}$ and shifted up 4 4units.

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$h\left(x\right)=\frac{1}{2}{x}^2+4$is given by:

Step 3. State the interpretation of the graph.

The function can be written as$f\left(x\right)=a{x}^2+c$, where$a=\frac{1}{2}$and$c=4$.

Note: When $c$is positive, the graph of a function is translated up and if $c$is negative, the graph of the function is translated down.

Since$0<a<1$and$c$is positive, the graph $h\left(x\right)=\frac{1}{2}{x}^2+4$is the graph$f\left(x\right)=x^2$vertically compressed by a factor$\frac{1}{2}$and shifted up 4 units.