Question

Determine whether the graphs of the following equations and functions have symmetry about the \(x\) -axis, the \(y\) -axis, or the origin. Check your work by graphing. $$|x|+|y|=1$$

Short answer

Question: Determine the symmetry of the graph of the equation \(|x|+|y|=1\). Answer: The graph is symmetric about the x-axis, y-axis, and the origin.

Step-by-step solution

Test for symmetry about the x-axis

To test for symmetry about the x-axis, replace \(y\) with \(-y\) in the equation and check if the equation remains unchanged: $$|x| + |-y| = 1$$ $$|x| + |y| = 1$$ The equation remains unchanged after the substitution, which indicates that the graph is symmetric about the x-axis.

Test for symmetry about the y-axis

To test for symmetry about the y-axis, replace \(x\) with \(-x\) in the equation and check if the equation remains unchanged: $$|-x| + |y| = 1$$ $$|x| + |y| = 1$$ The equation remains unchanged after the substitution, which indicates that the graph is symmetric about the y-axis.

Test for symmetry about the origin

To test for symmetry about the origin, replace both \(x\) and \(y\) with \(-x\) and \(-y\) in the equation and check if the equation remains unchanged: $$|-x| + |-y| = 1$$ $$|x| + |y| = 1$$ The equation remains unchanged after both substitutions, which indicates that the graph is also symmetric about the origin.

Conclusion

The graph of the equation \(|x|+|y|=1\) is symmetric about the \(x\)-axis, \(y\)-axis, and the origin. It is important to check the graph's symmetry to confirm these results.

Key concepts

Absolute Value Equations

Absolute value equations often involve expressions where an unknown variable lies inside an absolute value operator, such as \(|x|\). This operator measures the distance of a number from zero on a number line without considering its direction. Hence, whether the number is positive or negative, the outcome will always be positive.

For the equation \(|x| + |y| = 1\), both \(|x|\) and \(|y|\) are absolute values. This equation suggests a combination of two variables whose sum, when considered as absolute distances, equals 1. To visualize:

• Draw a coordinate plane where each grid step represents a consistent measure.
• The solution set of this equation will create a symmetrical shape on this plane.

The inherent nature of absolute values allows equations to be tested easily for symmetry, as computing negated versions of variables results in identical expression forms.

Graphical Symmetry Tests

Understanding the symmetry of a graph is crucial in fields such as engineering and physics where balance and equilibrium are often modeled graphically. By using graphical symmetry tests, we can identify an equation's inherent symmetrical properties.

The symmetry tests about different axes ensure if a graph looks identical when flipped or rotated in specific ways:

• **Symmetry about the x-axis**: Replace \(y\) with \(-y\). If the equation remains unchanged, any point \((x, y)\) on the graph should have a corresponding point \((x, -y)\).
• **Symmetry about the y-axis**: Replace \(x\) with \(-x\). Here, if untouched, each point \((x, y)\) is mirrored as \((-x, y)\).
• **Symmetry about the origin**: Replace both \(x\) and \(y\) with \(-x\) and \(-y\). Unchanged results indicate the point \((x, y)\) corresponds to \((-x, -y)\), implying identity through a 180-degree rotation.

Such tests reveal that the given equation demonstrates full symmetry about the x-axis, y-axis, and origin, as each substitution led to the same expression.

Coordinate Axes Symmetry

Symmetry concerning coordinate axes can simplify understanding the geometric appearance of equations. The graph of \(|x| + |y| = 1\) is an excellent demonstration of symmetry with respect to both axes:

• If symmetric about the **x-axis**, any alterations above and below this axis are reflections of each other.
• Symmetry about the **y-axis** implies left and right halves are direct mirrors.
• When respecting the **origin**, the graph maintains a centrally symmetric appearance in all quadrants.

The equation \(|x| + |y| = 1\) actually outlines a diamond shape, or a square rotated by 45 degrees, centered at the origin. Such symmetry assures that its form remains consistent across reflections and rotations, making it predictable and orderly in nature.