Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 41

Chapter 3, Problem 41

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. (-4, -2)

Verified step by step guidance

Identify the original point given as $(-4, -2)$. This point is located 4 units to the left of the origin on the x-axis and 2 units down on 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 $(-4, 2)$.

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 $(4, -2)$.

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 $(4, 2)$.

Plot all points on the coordinate plane: the original point $(-4, -2)$, the x-axis symmetric point $(-4, 2)$, the y-axis symmetric point $(4, -2)$, and the origin symmetric point $(4, 2)$ to visualize their relationships.

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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 is a two-dimensional surface defined by the x-axis (horizontal) and y-axis (vertical). Each point is represented by an ordered pair (x, y), where x indicates horizontal position and y indicates vertical position. Plotting a point involves locating its position based on these coordinates.

Symmetry with Respect to the Axes

Symmetry about the x-axis means reflecting a point across the x-axis, changing the y-coordinate's sign while keeping x the same. Symmetry about the y-axis involves reflecting across the y-axis, changing the x-coordinate's sign while keeping y the same. These reflections produce points that are mirror images across the respective axes.

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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