Consider the examples shown in the book. How are the graphs of g(x)=|x+3|-5 and f(x)=|x| related? How are the graphs of g(x)=|x-4|+2 and f(x)=|x| related?
See solution.
Practice makes perfect
To answer this question, we will consider the two examples shown in the book.
First Example
We will rewrite g(x)=|x+3|-5 in order to match g(x)=|x- h|+k.
g(x)=|x+3|-5 ⇔ g(x)=|x-( - 3)|+(- 5)
Let's now consider its graph.
We see that the graph of g(x)=|x-( - 3)|+(- 5) is a horizontal translation of three units to the left, and a vertical translation of five units down of the parent function f(x)=|x|.
Second Example
Let's now consider the graph of g(x)=|x- 4|+2.
We see that the graph of g(x)=|x- 4|+2 is a horizontal translation of four units to the right, and a vertical translation of two units up of the parent function f(x)=|x|.
Conclusion
Based on the examples, we can state that the graph of g(x)=|x- h|+k is a horizontal translation of h units and a vertical translation of k units. Also, consider the signs of h and k.
If h is greater than zero, the horizontal translation is done to the right. If h is less than zero, the horizontal translation is done to the left.
Similarly, if k is greater than zero, the vertical translation goes up. If k is less than zero, the vertical translation goes down.
