Math Problem Statement

Determine visually whether the graph is symmetric with respect to the​ x-axis, the​ y-axis, and the origin.

Question content area bottom left

Part 1

Is the graph symmetric with respect to the​ x-axis? Choose the correct answer below.

A.

​No; because when the graph is folded along the​ x-axis, the parts above and below the​ x-axis do not coincide.

B.

​Yes; because when the graph is folded along the​ x-axis, the parts above and below the​ x-axis coincide.

C.

​No; because when the graph is folded along the line

xequals=​y,

the parts above and below the line

xequals=y

do not coincide.

D.

​Yes; because when the graph is rotated

180 degrees180°

about the​ origin, the parts above and below the​ x-axis coincide.

...

Question content area right

Part 1

xy

Solution

To determine whether a graph is symmetric with respect to the x-axis, y-axis, or the origin, we need to analyze the graph based on certain properties:

  • Symmetry with respect to the x-axis: If a graph is symmetric with respect to the x-axis, the portion of the graph above the x-axis will be a mirror image of the portion below it.
  • Symmetry with respect to the y-axis: If a graph is symmetric with respect to the y-axis, the portion of the graph to the left of the y-axis will be a mirror image of the portion to the right.
  • Symmetry with respect to the origin: If a graph is symmetric with respect to the origin, rotating the graph 180 degrees about the origin will result in the same graph.

Steps to determine if the graph is symmetric about the x-axis:

  • Option B is correct if, when folded along the x-axis, the parts above and below the x-axis align perfectly (i.e., one side is a mirror image of the other).

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Math Problem Analysis

Mathematical Concepts

Graph Symmetry
Geometry
Coordinate Geometry

Formulas

-

Theorems

Symmetry with respect to the x-axis
Symmetry with respect to the y-axis
Symmetry with respect to the origin

Suitable Grade Level

Grades 8-10