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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 90
Chapter 2, Problem 90

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these.
∣y∣=−x |y| = -x

Verified step by step guidance
1
Rewrite the given equation: \(|y| = -x\).
Recall the symmetry tests: - For symmetry about the x-axis, replace \(y\) with \(-y\) and check if the equation remains unchanged. - For symmetry about the y-axis, replace \(x\) with \(-x\) and check if the equation remains unchanged. - For symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) and check if the equation remains unchanged.
Test for x-axis symmetry: Replace \(y\) with \(-y\) in \(|y| = -x\). Since \(|y| = |-y|\), the equation becomes \(|y| = -x\), which is the same as the original equation. So, the equation is symmetric with respect to the x-axis.
Test for y-axis symmetry: Replace \(x\) with \(-x\) in \(|y| = -x\). The equation becomes \(|y| = -(-x) = x\). This is not the same as the original equation, so it is not symmetric about the y-axis.
Test for origin symmetry: Replace \(x\) with \(-x\) and \(y\) with \(-y\). The equation becomes \(|-y| = -(-x)\), which simplifies to \(|y| = x\). This is not the same as the original equation, so it is not symmetric about the origin.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Symmetry in Graphs

Symmetry in graphs refers to how a graph mirrors itself across a line or point. Common symmetries include the x-axis, y-axis, and origin. Testing symmetry involves substituting variables (e.g., replacing y with -y for x-axis symmetry) and checking if the equation remains unchanged.
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Graphs and Coordinates - Example

Absolute Value Function

The absolute value function, denoted |y|, represents the non-negative value of y regardless of its sign. It affects the graph by reflecting negative y-values to positive, which influences symmetry and the shape of the graph.
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Domain and Range Restrictions

Understanding the domain and range is crucial, especially when absolute values and negative signs are involved. For example, |y| is always non-negative, so an equation like |y| = -x restricts x to values where the right side is non-negative, impacting the graph's existence and symmetry.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 3x2 - 14x | = 5

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Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | x2 + 5x + 5 | = 1

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Textbook Question

Simplify each power of i. i25

Textbook Question

Solve each equation. (x-1)2/3+(x-1)1/3 -12 = 0

Textbook Question

Simplify each power of i. i29

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Textbook Question

Solve each problem. A baseball is hit so that its height, s, in feet after t seconds is s=-16t2+44t+4. For what time period is the ball at least 32 ft above the ground?