How do you graph #g(x)=abs(x^2-4)# using transformations?

Question

Benjamin Achtenberg · Accepted Answer

See the graph and the explanation.

#y=|x^2-4| >=0# is the combined equation for the pair of parabolas #y=X^2-4, x ouside [-2, 2] and# #y=-(x^2-4), x in [-2, 2] .# Note that Only the parts of the parabolas for #y>=0 are included. Also, each is the mirror image of the missing part of the other, with respect to the x-axis. graph{y-|x^2-4|=0 [-10, 10, -5, 5]}

Lincoln Anderson · Answer

undefined To graph the function g(x) = |x^2 - 4| using transformations:

1. Start with the parent function y = |x|, which is a V-shaped graph centered at the origin.
2. Apply the transformation (x^2 - 4) inside the absolute value function. This shifts the graph horizontally by 4 units to the right and reflects it across the y-axis.
3. Since the coefficient of x^2 is positive, the graph opens upwards.
4. The vertex of the graph occurs where the expression inside the absolute value function equals zero, which is at x = ±2.
5. The graph is symmetric about the y-axis, so the portion of the graph for x < -2 is the reflection of the portion for x > 2.
6. Plot points on the graph to visualize the shape, especially around the vertex at (2, 0) and (-2, 0).

This process results in a graph that resembles a sideways "U" shape, with the vertex at (2, 0) and (-2, 0), opening upwards.