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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 86
Chapter 3, Problem 86

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. 5y2+5x2=305y^2 + 5x^2 =30

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Rewrite the given equation for clarity: \(5y^2 + 5x^2 = 30\).
To check symmetry with respect to the y-axis, replace \(x\) with \(-x\) and see if the equation remains unchanged: \(5y^2 + 5(-x)^2 = 30\).
To check symmetry with respect to the x-axis, replace \(y\) with \(-y\) and see if the equation remains unchanged: \(5(-y)^2 + 5x^2 = 30\).
To check symmetry with respect to the origin, replace both \(x\) with \(-x\) and \(y\) with \(-y\) and see if the equation remains unchanged: \(5(-y)^2 + 5(-x)^2 = 30\).
Compare the resulting equations from each substitution to the original equation to determine which symmetries hold true.

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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 the graph looks the same above and below the x-axis, reflecting points across it.
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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 indicates the graph mirrors itself on the left and right sides of the y-axis.
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07:42
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 produces the original equation. This means the graph is unchanged when rotated 180 degrees around the origin.
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Graph Hyperbolas NOT at the Origin
Related Practice
Textbook Question

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=√x,g(x)=1(x+5)ƒ(x)=\(\surd\) x,g(x)=\(\frac{1}{(x+5)}\)

Textbook Question

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

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, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√x, g(x)=3/(x+6)

Textbook Question

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

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Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=3√x-2

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