Textbook Question
Graph the line satisfying the given conditions. through (2, -4), m = 3/4
Ch. 2 - Graphs and Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 3, Problem 47
Without graphing, 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. See Examples 3 and 4. x2+y2=12
Verified step by step guidance1
Recall the tests for symmetry of a graph:
- Symmetry about the x-axis: Replace \(y\) with \(-y\) in the equation and see if the equation remains unchanged.
- Symmetry about the y-axis: Replace \(x\) with \(-x\) and check if the equation remains unchanged.
- Symmetry about the origin: Replace both \(x\) with \(-x\) and \(y\) with \(-y\) and check if the equation remains unchanged.
Start with the given equation:
\[x^2 + y^2 = 12\]
Test for symmetry about the x-axis by replacing \(y\) with \(-y\):
\[x^2 + (-y)^2 = 12\]
Simplify to get:
\[x^2 + y^2 = 12\]
Since the equation is unchanged, the graph is symmetric about the x-axis.
Test for symmetry about the y-axis by replacing \(x\) with \(-x\):
\[(-x)^2 + y^2 = 12\]
Simplify to get:
\[x^2 + y^2 = 12\]
Since the equation is unchanged, the graph is symmetric about the y-axis.
Test for symmetry about the origin by replacing \(x\) with \(-x\) and \(y\) with \(-y\):
\[(-x)^2 + (-y)^2 = 12\]
Simplify to get:
\[x^2 + y^2 = 12\]
Since the equation is unchanged, the graph is 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 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 for every point (x, y) on the graph, the point (x, -y) is also on the graph. Testing this helps identify if the graph mirrors across the x-axis.
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07:42
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. Checking this substitution reveals if the graph mirrors across the y-axis.
Recommended video:
07:42Properties 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 an equivalent equation. This means for every point (x, y), the point (-x, -y) is also on the graph. This test determines if the graph has rotational symmetry of 180 degrees about the origin.
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5:59Graph Hyperbolas NOT at the Origin
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