Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 39

Chapter 3, Problem 39

Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (5, -3)

Verified step by step guidance

Identify the given point as \( (5, -3) \). This means the point is located 5 units to the right of the origin along the x-axis and 3 units down along the y-axis.

To find the point symmetric with respect to the x-axis, keep the x-coordinate the same and change the sign of the y-coordinate. The symmetric point will be \( (5, 3) \).

To find the point symmetric with respect to the y-axis, keep the y-coordinate the same and change the sign of the x-coordinate. The symmetric point will be \( (-5, -3) \).

To find the point symmetric with respect to the origin, change the signs of both the x- and y-coordinates. The symmetric point will be \( (-5, 3) \).

Plot all points on the coordinate plane: the original point \( (5, -3) \), the x-axis symmetric point \( (5, 3) \), the y-axis symmetric point \( (-5, -3) \), and the origin symmetric point \( (-5, 3) \).

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Key Concepts

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

Coordinate Plane and Plotting Points

The coordinate plane consists of two perpendicular number lines called the x-axis and y-axis. Points are plotted using ordered pairs (x, y), where x indicates horizontal position and y indicates vertical position. Understanding how to locate and plot points is fundamental for visualizing symmetry.

Symmetry with Respect to the Axes

Symmetry about the x-axis means reflecting a point across the x-axis, changing the sign of the y-coordinate while keeping x the same. Symmetry about the y-axis involves changing the sign of the x-coordinate while keeping y the same. These reflections produce mirror images of the original point.

Symmetry with Respect to the Origin

Symmetry about the origin reflects a point through the origin, changing the signs of both coordinates. For a point (x, y), its symmetric point with respect to the origin is (-x, -y). This transformation is equivalent to a 180-degree rotation around the origin.

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