Exercise 7 Page 54

The given function is in the form of $g(x)=a\left|\frac{1}{b}(x-h)\right|+k,$ where $(h,k)$ is the vertex, $a$ is a parameter for vertical stretch/compression, and $b$ is a parameter for horizontal stretch/compression. We see above that $a$ $=$ $\frac{4}{3},$ $b$ $=$ $1,$ $h$ $=$ $5,$ and $k$ $=$ $7.$ Then, the vertex of $g(x)$ is at $(5,7),$ which means that the parent function is translated $5$ units right and $7$ units up.

$$\begin{array}{c}(0,0)\to (5,7)\end{array}$$

Since $a>1,$ then $g(x),$ in addition of being a translation, is also a vertical stretch of the parent function by a factor of $\frac{4}{3}.$ The $x\text{-}$coordinate of each point on the graph of the parent function will be shifted $5$ units to the right, and the $y\text{-}$coordinate will be stretched by a factor of $\frac{4}{3}$ and then moved up $7$ units. Let's consider the points $(\text{-}3,3)$ and $(3,3).$

$$\begin{array}{rl}(\text{-}3,3)& \to \left(\text{-}3+5,\frac{4}{3}|3|+7\right)\to (2,11)\\ (3,3)& \to \left(3+5,\frac{4}{3}|3|+7\right)\to (8,11)\end{array}$$

Now, we will plot the vertex and the above points, and graph $g(x).$ Recall that the graph of an absolute value function has a V-shape!

We see above that there are no restrictions for the values that the $x\text{-}$variable can take. Moreover, we also see that the $y\text{-}$variable takes values that are greater than or equal to $7.$ We will use this information to write the domain and range of the function.

$$\begin{array}{rl}\text{Domain:}& \text{all real numbers}\\ \text{Range:}& y\ge 7\end{array}$$