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

1. 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.
2. 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 \((-4, 2)\).
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 \((4, -2)\).
4. 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 \((4, 2)\).
5. 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.

Key Concepts

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.