Math Problem Statement
(c) If the graph is that of a function, determine what kinds of symmetry it has. Select all that apply. A. It is symmetric with respect to the y-axis. B. It is symmetric with respect to the origin. C. It is symmetric with respect to the x-axis. D. The graph is not symmetric with respect to the x-axis, y-axis, or the origin. E. The graph is not a function.
Solution
The graph shows a parabola, which appears to be symmetric with respect to the y-axis. We can determine this by observing that the graph is a U-shape centered at , meaning the left and right sides of the parabola are mirror images across the vertical line passing through .
Symmetry:
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A. Symmetric with respect to the y-axis: No, because the parabola is not centered at (the y-axis); it is centered at .
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B. Symmetric with respect to the origin: No, the graph does not have origin symmetry (it is not reflected through the origin).
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C. Symmetric with respect to the x-axis: No, parabolas generally do not have symmetry across the x-axis unless they are horizontal parabolas, which this is not.
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D. The graph is not symmetric with respect to the x-axis, y-axis, or the origin: Correct, this graph is not symmetric about these axes or the origin.
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E. The graph is not a function: No, the graph represents a parabola, which is a function. The graph passes the vertical line test, meaning it is indeed a function.
Conclusion:
The correct answer is D: The graph is not symmetric with respect to the x-axis, y-axis, or the origin.
Would you like more details on the concept of symmetry or have any other questions?
Related questions:
- What is the general form of the equation of a parabola with symmetry around a different vertical line?
- How do you check for symmetry with respect to the y-axis?
- Can a function be symmetric with respect to the origin and still be a function?
- What are the characteristics of parabolas that are symmetric about the x-axis?
- How does the vertical line test help to determine if a graph represents a function?
Tip: The symmetry of a function can be quickly identified by its equation. For example, a quadratic function is symmetric with respect to a vertical line determined by its vertex.
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Math Problem Analysis
Mathematical Concepts
Graph Symmetry
Quadratic Functions
Parabolas
Formulas
Equation of a parabola: y = a(x-h)^2 + k
Theorems
Vertical Line Test
Symmetry in Quadratic Functions
Suitable Grade Level
Grades 9-11