Q9. - Algebra 1

Question

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

 $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{3}{\mathit{x}}^{\mathbf{2}}$

Step-by-Step Solution

Answer

The graph $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{3}{\mathit{x}}^{\mathbf{2}}$ is the graph of $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$ reflected across the  $\mathit{x}$– axis and vertically stretched by a factor 3.

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 $g\left(x\right)=-3x^2$ 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. The coefficient of $x^2$ term is negative, so the graph is reflected across the $x$– axis. Since $a=3$, the graph is vertically stretched.

Therefore, the graph of $g\left(x\right)=-3x^2$ is the graph of $f\left(x\right)=x^2$ reflected across the $x$– axis and vertically stretched by a factor 3.