Question

Match the graph to one of the following functions:

(a) $f\left(x\right)=x^2+4$

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

(c) $f\left(x\right)={\left(x+4\right)}^2$

(d) $f\left(x\right)={\left(x-4\right)}^2$

(e) $f\left(x\right)={\left(x+4\right)}^2-4$

(f) $f\left(x\right)={\left(x+4\right)}^2+4$

(g)$f\left(x\right)={\left(x-4\right)}^2-4$

(h)$f\left(x\right)={\left(x-4\right)}^2+4$

Short answer

The given graph matches with the function (d)$f\left(x\right)={\left(x-4\right)}^2$

Step-by-step solution

Step 1. Given Information

A graph is given as

The given graph is of a quadratic function.

Consider the general form of a quadratic function.

$f\left(x\right)=a{\left(x-h\right)}^2+k$

Where the vertex is$\left(h,k\right)$

Step 2. Explanation

From the graph, the vertex is at $\left(4,0\right)$.

So,$\left(h,k\right)=\left(4,0\right)$

$h=4,k=0$

Now, find $a$by using another point on the graph.

Step 3. Value ofa

The point $\left(2,4\right)$is on the graph.

So, $f\left(2\right)=4$

$$\begin{gathered} a(2-4{)}^2+0=4 \\ a(-2{)}^2=4 \\ 4a=4 \\ a=1 \end{gathered}$$

So, the required function is $f\left(x\right)={\left(x-4\right)}^2$

The given graph matches with function (d).