Question

Plot each point. Then plot the point that is symmetric to it with respect to (a) the \(x\) -axis; (b) the y-axis; (c) the origin. $$ (0,-3) $$

Short answer

(a): (0, 3), (b): (0, -3), (c): (0, 3)

Step-by-step solution

Plot the original point

Start by plotting the point (0, -3) on a Cartesian coordinate system. This point is located on the y-axis, 3 units down from the origin.

Symmetric point with respect to the x-axis

To find the point symmetric to (0, -3) with respect to the x-axis, reflect the point over the x-axis. The new point will have the same x-coordinate but the opposite y-coordinate. So, the symmetric point is (0, 3).

Symmetric point with respect to the y-axis

To find the point symmetric to (0, -3) with respect to the y-axis, reflect the point over the y-axis. The new point will have the same y-coordinate but the opposite x-coordinate. Since the x-coordinate is already 0, the symmetric point remains the same at (0, -3).

Symmetric point with respect to the origin

To find the point symmetric to (0, -3) with respect to the origin, reflect the point over both the x-axis and y-axis. This means both the x and y-coordinates change signs. Therefore, the symmetric point is (0, 3).

Key concepts

Plotting Points

Understanding how to plot points on a Cartesian coordinate system is crucial.
Cartesian plane has two axes: the x-axis (horizontal) and the y-axis (vertical). Each point is defined by a pair of coordinates (x, y).
To plot a point, begin at the origin (where both axes meet, (0,0)) and move according to the coordinates:

• The first number in the pair is the x-coordinate. Move left (if it is negative) or right (if positive) along the x-axis.
• The second number is the y-coordinate. Move down (if negative) or up (if positive) along y-axis.

For instance, to plot the point (0, -3) on the Cartesian plane, start at the origin:

• Since the x-coordinate is 0, you do not move left or right.
• Then, move down 3 units on the y-axis as the y-coordinate is -3.

Now, mark that spot. You've plotted the point (0, -3)!

Reflection Over x-axis

Reflecting a point over the x-axis changes its y-coordinate to its opposite:
This process flips the point across the x-axis, keeping the x-coordinate the same.
For example, reflecting the point (0, -3) over the x-axis results in the point (0, 3) since:

• The x-coordinate stays 0.
• The y-coordinate changes from -3 to 3.

To visualize, imagine folding the coordinate plane along the x-axis. The point originally below the x-axis at (0, -3) will appear above the x-axis at (0, 3) after reflection.
This method is handy for recognizing symmetries and patterns in various mathematical contexts.

Reflection Over y-axis

Reflection over the y-axis changes the x-coordinate of a point to its opposite:
This process flips the point across the y-axis, keeping the y-coordinate the same.
Reflecting the point (0, -3) over the y-axis keeps the point unchanged at (0, -3) since:

• The y-coordinate stays -3.
• The x-coordinate is already 0, so it remains 0.

For visualization, imagine folding the coordinate plane along the y-axis. In this specific case, the point lies on the y-axis, so it doesn't change its position. This reflection process can help in understanding the symmetry across the vertical axis.

Reflection Over Origin

Reflection over the origin changes both the x and y-coordinates to their opposites.
This flips the point across both x and y-axes, essentially rotating it 180 degrees around the origin.
Reflecting the point (0, -3) over the origin results in the point (0, 3) since:

• The x-coordinate stays 0.
• The y-coordinate changes from -3 to 3.

To visualize, imagine rotating the point (0, -3) around the origin. The point (0, -3) moves to (0, 3) since both coordinates change signs. Understanding this reflection helps enhance comprehension of coordinate transformations and rotations in space.