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Sketch a graph of an even function that has a vertical asymptote at x = 2, a hole at x = −4 and as x → [infinity], y → 3.

User Heartpunk
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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.

Sketch a graph of an even function that has a vertical asymptote at x = 2, a hole-example-1
User Vinessa
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