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.

If the graph is that of a function, determine what kinds of symmetry it has. Select all that apply.

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 x=4x = -4, meaning the left and right sides of the parabola are mirror images across the vertical line passing through x=4x = -4.

Symmetry:

  • A. Symmetric with respect to the y-axis: No, because the parabola is not centered at x=0x = 0 (the y-axis); it is centered at x=4x = -4.

  • B. Symmetric with respect to the origin: No, the graph does not have origin symmetry (it is not reflected through the origin).

  • 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.

  • 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.

  • 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:

  1. What is the general form of the equation of a parabola with symmetry around a different vertical line?
  2. How do you check for symmetry with respect to the y-axis?
  3. Can a function be symmetric with respect to the origin and still be a function?
  4. What are the characteristics of parabolas that are symmetric about the x-axis?
  5. 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 y=ax2+bx+cy = ax^2 + bx + c 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