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Chapter 2: Problem 37

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. $$ (-3,-4) $$

Short Answer

Expert verified
Symmetric points are \((-3, 4)\), \((3, -4)\), and \((3, 4)\).

Step by step solution

01

- Plot the Given Point

Begin by plotting the point \((-3, -4)\) on the coordinate plane. This means moving 3 units to the left of the origin and 4 units down.
02

- Find the Symmetric Point with Respect to the x-axis

To find the symmetric point with respect to the x-axis, keep the x-coordinate the same and change the sign of the y-coordinate. Thus, the symmetric point is \((-3, 4)\). Plot this point.
03

- Find the Symmetric Point with Respect to the y-axis

To find the symmetric point with respect to the y-axis, keep the y-coordinate the same and change the sign of the x-coordinate. Thus, the symmetric point is \((3, -4)\). Plot this point.
04

- Find the Symmetric Point with Respect to the Origin

To find the symmetric point with respect to the origin, change the sign of both the x-coordinate and the y-coordinate. Thus, the symmetric point is \((3, 4)\). Plot this point.
05

- Review and Confirmation

Review all points plotted: the original point \((-3, -4)\), symmetric to the x-axis \((-3, 4)\), symmetric to the y-axis \((3, -4)\), and symmetric to the origin \((3, 4)\). Ensure all points are accurately plotted.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Points
Plotting points is a fundamental skill in coordinate geometry. Each point on the plane is defined by an ordered pair \((x, y)\). The first number is the x-coordinate, which tells us how far left or right to move from the origin (0,0). The second number is the y-coordinate, which tells us how far up or down to move.

For instance, to plot the point \((-3, -4)\), move 3 units left (since x is -3) and 4 units down (since y is -4). This visual placement helps in understanding the spatial relationships between points.
Coordinate Plane
The coordinate plane is a two-dimensional surface formed by two perpendicular number lines: the x-axis and the y-axis. They intersect at the origin (0,0).

Each point on the plane is located using its x and y coordinates. Positive x-values are to the right of the origin, and negative x-values are to the left. For y-values, positives are above the origin, and negatives are below. This grid system simplifies the process of locating points and visualizing geometric concepts like symmetry.
Symmetry with Respect to Axes
Symmetry in coordinate geometry is about reflection. When finding a point symmetric with respect to the x-axis, the x-coordinate remains unchanged, but the y-coordinate changes sign. So, for the point \((-3, -4)\), its x-axis symmetric point would be \((-3, 4)\).

Similarly, for symmetry with respect to the y-axis, the y-coordinate remains the same while the x-coordinate changes sign. Thus, from \((-3, -4)\) to its y-axis symmetric point \((3, -4)\). Understanding these reflections helps in graphing techniques and solving geometry problems.
Symmetry with Respect to Origin
Symmetry with respect to the origin means reflecting a point through the (0,0) point. Both coordinates of the point change signs. For example, the point \((-3, -4)\) becomes \((3, 4)\) when reflected through the origin.

This concept is essentially about flipping both the x and y values to get a point diagonally opposite on the coordinate plane. Grasping these reflections makes understanding the geometry and plotting much more intuitive.

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