Graph f(x)=|x^2-4|

Problem

ƒ(x)=|x2−4|

Solution

1. Identify the base function inside the absolute value, which is the parabola y=x2−4

2. Find the intercepts of the base function by setting x2−4=0 which gives xintercepts at (−2,0) and (2,0) and a yintercept at (0,−4)

3. Determine the behavior of the absolute value, which reflects any part of the graph below the xaxis (y<0 across the xaxis to make it positive.

4. Apply the reflection to the interval (−2,2) where x2−4 is negative; the vertex (0,−4) moves to (0,4)

5. Combine the parts to form the final graph: the original parabola for x≤−2 and x≥2 and the reflected upward curve for −2<x<2

Final Answer

ƒ(x)={[x2−4,if *|x|≥2],[−(x2−4),if *|x|<2])

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