Question
Which of the following graphs represents an even function?
(1 point)
Responses
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.
Image with alt text: 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.
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(1 point)
Responses
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.
Image with alt text: 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.
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Answers
GPT-4o mini
An even function is defined as a function that satisfies the condition \( f(-x) = f(x) \) for all \( x \). This means that the graph of an even function is symmetric about the y-axis.
Looking at the descriptions of the graphs provided:
1. **First graph**: The curve decreases through the origin, suggesting it does not exhibit y-axis symmetry as it behaves differently on either side of the y-axis.
2. **Second graph**: The presence of cusps at the origin indicates it may not exhibit symmetry in the same way as an even function does. This graph also does not maintain the required symmetry about the y-axis.
3. **Third graph**: This graph has multiple curves and asymptotes, indicating it does not maintain y-axis symmetry. The behavior of the function is different for negative and positive values of \( x \).
4. **Fourth graph**: This is described as a parabola that opens to the right with its vertex at the origin. A parabola that opens sideways is not classified as even since even functions must be symmetric about the y-axis.
None of the graphs definitively describe an even function, as all fail the criterion of symmetry about the y-axis. If you had to choose the closest representation, one that might still retain some features of symmetry near the origin would be the parabola, but since the parabolaOpening rightward does not satisfy the definition of an even function, based on the information provided, none of these graphs are even functions.
Looking at the descriptions of the graphs provided:
1. **First graph**: The curve decreases through the origin, suggesting it does not exhibit y-axis symmetry as it behaves differently on either side of the y-axis.
2. **Second graph**: The presence of cusps at the origin indicates it may not exhibit symmetry in the same way as an even function does. This graph also does not maintain the required symmetry about the y-axis.
3. **Third graph**: This graph has multiple curves and asymptotes, indicating it does not maintain y-axis symmetry. The behavior of the function is different for negative and positive values of \( x \).
4. **Fourth graph**: This is described as a parabola that opens to the right with its vertex at the origin. A parabola that opens sideways is not classified as even since even functions must be symmetric about the y-axis.
None of the graphs definitively describe an even function, as all fail the criterion of symmetry about the y-axis. If you had to choose the closest representation, one that might still retain some features of symmetry near the origin would be the parabola, but since the parabolaOpening rightward does not satisfy the definition of an even function, based on the information provided, none of these graphs are even functions.