Chapter 8.4, Problem 46E

To determine

To match: The correct graph for given function $r(x)=f(x+2),\text{where}f(x)={\left(x-2\right)}^2+1$

Answer to Problem 46E

The graph of $r(x)=f(x+2)$ is match with $C$ .

Explanation of Solution

Given information :

Following graphs provided

  

As we know that the vertex of the graph of $f(x)=m{(x-h)}^2+k\left(\text{where}m\text{is}\text{constant}\right)$ is $(h,k)$ and the axis of symmetry is $x=h$ .

Now compare given function $f(x)={\left(x-2\right)}^2+1$ with $f(x)=m{(x-h)}^2+k$ , we get $h=2,k=1$

Thus, the vertex of $f(x)$ is $(2,1)$ and the axis of symmetry $x=2$

Since $r(x)=f(x+2)$ , here $h=-2$ , so the graph of $r(x)$ is a horizontal translation of $2$ units left of the graph of $f(x)$ ,

So the vertex of $r(x)$ is $(0,1)$ and the axis of symmetry $x=0.$

Thus, the graph $C$ has a vertex $(0,1)$ and the axis of symmetry $x=0.$

Hence, the graph of $r(x)=f(x+2)$ is match with $C$ .