Question

(a) If a graph is symmetric with respect to the $x$ -axis and $(a, b)$ is on the graph, then______is also on the graph. (b) If a graph is symmetric with respect to the $y$ -axis and $(a, b)$ is on the graph, then_ is also on the graph. (c) If a graph is symmetric about the origin and $(a, b)$ is on the graph, then____________ is also on the graph.

Name: Problem 7, Chapter 1: Precalculus Mathematics for Calculus
Uploaded: 2020-06-04T23:04:40-08:00
Duration: 2 min 17 s
Channel: Abigail Martyr
Description: (a) If a graph is symmetric with respect to the x -axis and (a, b) is on the graph, thenis also on the graph. (b) If a graph is symmetric with respect to the y -axis and (a, b) is on the graph, then is also on the graph. (c) If a graph is symmetric about the origin and (a, b) is on the graph, then is also on the graph.

   (a) If a graph is symmetric with respect to the $x$ -axis and
$(a, b)$ is on the graph, then____________is also on the
graph.

(b) If a graph is symmetric with respect to the $y$ -axis and
$(a, b)$ is on the graph, then___________ is also on the
graph.
(c) If a graph is symmetric about the origin and $(a, b)$ is on
the graph, then________________ is also on the graph.

Precalculus Mathematics for Calculus

James Stewart,… 7th Edition

Chapter 1, Problem 7

Instant Answer

Solved by Expert Abigail Martyr

06/04/2020

Step 1

If a point $(a, b)$ is on the graph, then there should be a corresponding point that is the same distance from the $x$-axis but on the opposite side. This point would have the same $x$-value but the $y$-value would be the negative of the original point's Show more…

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(a) If a graph is symmetric with respect to the $x$ -axis and $(a, b)$ is on the graph, then____________is also on the graph. (b) If a graph is symmetric with respect to the $y$ -axis and $(a, b)$ is on the graph, then___________ is also on the graph. (c) If a graph is symmetric about the origin and $(a, b)$ is on the graph, then________________ is also on the graph.

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Key Concepts

Reflection Symmetry with respect to the x-axis

This concept means that the graph remains unchanged when reflected over the x-axis. For any point (a, b) on the graph, its mirror image with respect to the x-axis is (a, -b), meaning the sign of the y-coordinate is reversed while the x-coordinate remains the same.

Reflection Symmetry with respect to the y-axis

This concept indicates that the graph is unchanged under reflection over the y-axis. For any point (a, b) on the graph, its mirror image with respect to the y-axis is (-a, b), which involves changing the sign of the x-coordinate while keeping the y-coordinate unchanged.

Rotational Symmetry about the Origin

Also known as origin symmetry, this concept means that the graph remains invariant when rotated 180 degrees about the origin. For any point (a, b) on the graph, its corresponding point under this rotation is (-a, -b), where both the x-coordinate and the y-coordinate change signs.

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