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

1. Use the vertex y-value: k = −5 (since vertex is (h, −5)).
2. Plug a given point into f(x)=|x−h|+k: for (−6, −2), |−6−h|−5 = −2 so |−6−h| = 3. The same holds for x=0, so the two x-values where y=−2 are h±3. Those x-values are −6 and 0, so h is the midpoint: h = (−6+0)/2 = −3. Algebraically you also get h = −3 from solving h±3 = {−6,0}.

Thus h = −3 and f(x)=|x+3|−5.

Translation: compared to the parent y=|x|, the graph is shifted 3 units left (because h = −3) and 5 units down (k = −5).