Exercise 1 Page 66

We are given the graphs of two functions, namely, $f(x)=x^2$ and $g(x).$

First, we see that the vertex of $g(x)$ is $2$ units above the vertex of $f(x).$ We will start by translating the graph of $f(x)$ up $2$ units. Now, let's plot $g(x)$ and $h(x)$ on the same coordinate plane and compare them.

We see that the graphs of $g(x)$ and $h(x)$ still do not match. However, we can see that the graph of $g(x)$ is wider, which means that we have to make a horizontal stretch. In other words, $g(x)=h(ax).$

To figure out the scale factor, we will use the fact that $g(2)=3.$ Next, we substitute $h(x)=x^2+2$ and solve the equation for $a.$ Since $a$ must be positive, we will discard the negative option. Now, we are ready to write our function $g(x)$ in terms of $f(x).$ In conclusion, to obtain $g(x)$ we must translate the graph of $f(x)$ $2$ units up and then stretch it horizontally by a factor of $\frac{1}{2}.$