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. Step 1: Start by plotting the given point (-4, -2) on the Cartesian plane. This point is located 4 units to the left of the origin (because of the negative x-coordinate) and 2 units below the origin (because of the negative y-coordinate).
2. Step 2: To find the point that is symmetric to (-4, -2) with respect to the x-axis, you need to change the sign of the y-coordinate while keeping the x-coordinate the same. This is because when we reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate changes sign. So, the point symmetric to (-4, -2) with respect to the x-axis is (-4, 2). Plot this point on the same Cartesian plane.
3. Step 3: To find the point that is symmetric to (-4, -2) with respect to the y-axis, you need to change the sign of the x-coordinate while keeping the y-coordinate the same. This is because when we reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate changes sign. So, the point symmetric to (-4, -2) with respect to the y-axis is (4, -2). Plot this point on the same Cartesian plane.
4. Step 4: To find the point that is symmetric to (-4, -2) with respect to the origin, you need to change the signs of both the x-coordinate and the y-coordinate. This is because when we reflect a point across the origin, both the x-coordinate and the y-coordinate change signs. So, the point symmetric to (-4, -2) with respect to the origin is (4, 2). Plot this point on the same Cartesian plane.
5. Step 5: Now, you should have four points plotted: the original point (-4, -2), and its reflections across the x-axis (-4, 2), the y-axis (4, -2), and the origin (4, 2).

Key Concepts

Symmetry with respect to the x-axis

A point (x, y) is symmetric to the x-axis if its reflection across the x-axis is (x, -y). This means that the x-coordinate remains the same while the y-coordinate changes sign. For example, the point (-4, -2) would reflect to (-4, 2) when considering symmetry about the x-axis.

Symmetry with respect to the y-axis

A point (x, y) is symmetric to the y-axis if its reflection across the y-axis is (-x, y). In this case, the y-coordinate remains unchanged while the x-coordinate changes sign. For the point (-4, -2), the symmetric point with respect to the y-axis would be (4, -2).

Symmetry with respect to the origin

A point (x, y) is symmetric to the origin if its reflection through the origin is (-x, -y). This transformation involves changing the signs of both coordinates. For the point (-4, -2), the symmetric point with respect to the origin would be (4, 2).