Question

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

$$\begin{gathered} f\left(x\right)=x^2 \\ h\left(x\right)=2x^2 \end{gathered}$$

Short answer

The graph of $h\left(x\right)=2x^2$ is the graph of $f\left(x\right)=x^2$,vertically stretched by a factor 2.

Step-by-step solution

Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where,$a\ne 0$ is called the standard form of the quadratic function.

Step 2. Define the transformations of the graph of the function.

The graph of $kf\left(x\right)$, will vertically stretch the graph of $f\left(x\right)$ by a factor $k$ if$k>1$ and will vertically compress the graph of $f\left(x\right)$ by a factor $k$ if $0<k<1$.

Step 3. Describe how the graph of hx=2x2 is related to the graph of fx=x2.

Observe the equation $h\left(x\right)=2x^2$.

The graph is multiplied by $k=2$, so the graph is vertically stretched by a factor 2.

Therefore, the graph of $h\left(x\right)=2x^2$is the graph of $f\left(x\right)=x^2$, vertically stretched by a factor 2.