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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 88
Chapter 3, Problem 88

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.
y3=x+4y^3 = x + 4

Verified step by step guidance
1
Rewrite the given equation: \(y^3 = x + 4\).
To test symmetry about the y-axis, replace \(x\) with \(-x\) and check if the equation remains unchanged: \(y^3 = -x + 4\).
To test symmetry about the x-axis, replace \(y\) with \(-y\) and check if the equation remains unchanged: \((-y)^3 = x + 4\), which simplifies to \(-y^3 = x + 4\).
To test symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) simultaneously and check if the equation remains unchanged: \((-y)^3 = -x + 4\), which simplifies to \(-y^3 = -x + 4\).
Compare the transformed equations to the original to determine which symmetry (x-axis, y-axis, origin) the graph has, or if it has none.

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Key Concepts

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

Symmetry with Respect to the x-axis

A graph is symmetric about the x-axis if replacing y with -y in the equation yields an equivalent equation. This means that for every point (x, y) on the graph, the point (x, -y) is also on the graph, reflecting the graph across the x-axis.
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Properties of Parabolas

Symmetry with Respect to the y-axis

A graph is symmetric about the y-axis if replacing x with -x in the equation results in the same equation. This implies that for every point (x, y), the point (-x, y) is also on the graph, reflecting the graph across the y-axis.
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Properties of Parabolas

Symmetry with Respect to the Origin

A graph is symmetric about the origin if replacing both x with -x and y with -y in the equation produces an equivalent equation. This means that for every point (x, y), the point (-x, -y) is also on the graph, indicating rotational symmetry of 180 degrees about the origin.
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Graph Hyperbolas NOT at the Origin
Related Practice
Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=1/(x+4), g(x)=-(1/x)

Textbook Question

Determine the largest open intervals of the domain over which each function is (c) constant. See Example 9.

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Textbook Question

Determine the largest open intervals of the domain over which each function is b) decreasing. See Example 9.

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Textbook Question

Given functions f and g, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

Textbook Question

What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

Textbook Question

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. |x| = |y|

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