Final answer:
An even function with a vertical asymptote at x = 2 and a hole at x = -4, leveling off to y = 3 as x approaches infinity, should have corresponding symmetry about the y-axis with asymptotes and holes reflected on both sides due to its even nature.
Step-by-step explanation:
To sketch a graph of an even function with the specified characteristics, we must first understand what an even function is.
An even function is symmetric about the y-axis, meaning that for any point (x, y) on the graph of the function, the point (-x, y) will also be on the graph.
This symmetry will help us visualize the function around the given points.
A vertical asymptote at x = 2 suggests that as the function approaches x = 2 from either side, the function values grow without bound.
However, since it is an even function, we must also have a vertical asymptote at x = -2 due to the symmetry.
A hole at x = -4 indicates that the function is not defined at that particular x-value, but it can approach a certain y-value as x gets closer to -4 from both sides.
Given that as x approaches infinity, y approaches 3, we know that the graph will level off towards the horizontal line y = 3 at both ends, creating a horizontal asymptote.
When sketching, one would begin by drawing the vertical asymptotes at x = 2 and x = -2 due to even symmetry.
Next, a small circle or dot can represent the hole at x = -4, and due to symmetry, another hole would exist at x = 4.
Then, we sketch the function levelling off to y = 3 as x approaches both positive and negative infinity.
The final sketch should reflect these features while ensuring the graph remains symmetric about the y-axis.