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Which of the following graphs represents an even function?

(1 point)
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A graph is shown on the coordinate plane. The graph is a curve that increases from negative infinity to the point left parenthesis negative 1.095 comma 1.051 right parenthesis, then decreases through the origin to point left parenthesis 1.095 comma negative 1.051 right parenthesis and then increases to infinity.
Image with alt text: A graph is shown on the coordinate plane. The graph is a curve that increases from negative infinity to the point left parenthesis negative 1.095 comma 1.051 right parenthesis, then decreases through the origin to point left parenthesis 1.095 comma negative 1.051 right parenthesis and then increases to infinity.

A graph is shown on the coordinate plane.The graph is a continuous curve that decreases from infinity to a cusp at the point left parenthesis negative 1.316 comma 0 right parenthesis, then increases to the point left parenthesis negative 0.88 comma 2.112 right parenthesis, then decreases to a cusp at the origin, then increases to the point left parenthesis 0.88 comma 2.112 right parenthesis, then decreases to a cusp at the point left parenthesis 1.316 comma 0 right parenthesis, then increases to infinity.
Image with alt text: A graph is shown on the coordinate plane. The graph is a continuous curve that decreases from infinity to a cusp at the point left parenthesis negative 1.316 comma 0 right parenthesis, then increases to the point left parenthesis negative 0.88 comma 2.112 right parenthesis, then decreases to a cusp at the origin, then increases to the point left parenthesis 0.88 comma 2.112 right parenthesis, then decreases to a cusp at the point left parenthesis 1.316 comma 0 right parenthesis, then increases to infinity.

A graph is shown on the coordinate plane.The graph is a rational function consisting of three curves. The first curve increases from negative infinity to the point left parenthesis negative 0.684 comma negative 1.223 right parenthesis then decreases to negative infinity. The second curve decreases from infinity to the point left parenthesis 0.628 comma 1.789 right parenthesis then increases to infinity. The third curve increases from negative infinity to positive infinity. The graph has asymptotes at x = 0, x = 2, and y = x.
Image with alt text: A graph is shown on the coordinate plane. The graph is a rational function consisting of three curves. The first curve increases from negative infinity to the point left parenthesis negative 0.684 comma negative 1.223 right parenthesis then decreases to negative infinity. The second curve decreases from infinity to the point left parenthesis 0.628 comma 1.789 right parenthesis then increases to infinity. The third curve increases from negative infinity to positive infinity. The graph has asymptotes at x = 0, x = 2, and y = x.

A graph is shown on the coordinate plane. The graph is a parabola that opens right with a vertex at left parenthesis 0 comma 0 right parenthesis.

1 answer

To determine which graph represents an even function, remember that a function is considered even if it satisfies the property \( f(-x) = f(x) \) for all \( x \). Geometrically, this means that the graph of the function is symmetric about the y-axis.

Analyzing each of the provided descriptions:

  1. Graph 1: Decreases through the origin and shows no symmetric behavior about the y-axis. (Not even)

  2. Graph 2: Features a cusp at the origin and symmetric behavior (the left side mirrors the right side). (Even)

  3. Graph 3: Describes a rational function with three curves, but the overall shape suggests it lacks symmetry about the y-axis. (Not even)

  4. Graph 4: A parabola that opens right with a vertex at the origin does not exhibit symmetry about the y-axis in the standard sense (parabolas open either up or down for even functions). (Not even)

Given these observations, Graph 2 is the only graph that exhibits the necessary symmetry about the y-axis and thus represents an even function.