Question

In Problems 7–18, match each graph to one of the following functions:

$$\begin{gathered} \mathit{A}\mathbf{.}y=x^2+2\mathit{B}\mathbf{.}y=-x^2+2\mathit{C}.y=\left|x\right|+2 \\ \mathit{D}\mathbf{.}y=-\left|x\right|+2\mathit{E}\mathbf{.}y={(x-2)}^2\mathit{F}\mathbf{.}y=-{(x+2)}^2 \\ \mathit{G}\mathbf{.}y=\left|x-2\right|\mathit{H}\mathbf{.}y=-\left|x+2\right|\mathit{I}\mathbf{.}y=2x^2 \\ \mathit{J}\mathbf{.}y=-2x^2\mathit{K}\mathbf{.}y=2\left|x\right|\mathit{L}\mathbf{.}y=-2\left|x\right| \end{gathered}$$

Short answer

The given graph is matched to the function$\mathit{D}\mathbf{.}\mathbf{}y=-\left|x\right|+2.$

Step-by-step solution

Step 1. Given Information 

There is the given graph we have to match each graph to its functions.

Step 2. Identifying the function of the graph 

When the right side of the function $y=f\left(x\right)$is multiplied by $-1$the graph of the new function $y=-f\left(x\right)$is the reflection about the x-axis of the graph of the function $y=f\left(x\right)$.

Thus, the function whose graph is the reflection of the graph of $y=\left|x\right|$in the x-axis is$y=-\left|x\right|.$

Step 3. Stating the function of the graph 

If a positive real number k is added to the outputs of a function $y=f\left(x\right),$the

graph of the new function $y=f\left(x\right)+k$ is the graph of fshifted vertically up

kunits.

Since the graph of $y=-\left|x\right|$is shifted vertically up by $2$units.

The function of the given graph is$y=-\left|x\right|+2$that matched to the option D.