Question

The graph of f(x) = |x – h| + k contains the points (–6, –2) and (0, –2). The graph has a vertex at (h, –5). Describe how to find the value of h. Then, explain how this value translates the graph of the parent function.

Answer 1

To find the value of h, we first need to use the given vertex information. The vertex of the absolute value function f(x) = |x – h| + k is given as (h, -5). Since the y-coordinate of the vertex is -5, we have k = -5.

Next, we use the information about the points (–6, –2) and (0, –2) to find the value of h. We know that when x = -6, f(x) = -2. So, substituting these values into the function gives us:

|-6 – h| - 5 = -2
|6 + h| = 3

Similarly, when x = 0, f(x) = -2. So,

|0 - h| - 5 = -2
|h| - 5 = -2
|h| = 3

By combining the two equations, we can find the value of h:

|h| = 3
h = 3 or h = -3

Therefore, the value of h can be either 3 or -3.

Translating the graph of the parent function f(x) = |x| can be thought of as shifting the graph horizontally. If h = 3, the graph will shift 3 units to the right, and if h = -3, the graph will shift 3 units to the left. This results in the horizontal translation of the graph of f(x) = |x – h| + k.