Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (a) x-axis (-4, -2)

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Start by plotting the original point \((-4, -2)\) on the coordinate plane. This means moving 4 units to the left of the origin (because of the negative x-value) and 2 units down (because of the negative y-value).

To find the point symmetric to \((-4, -2)\) with respect to the x-axis, recall that reflecting a point over the x-axis changes the sign of the y-coordinate but keeps the x-coordinate the same.

Apply this rule: keep the x-coordinate as \(-4\), and change the y-coordinate from \(-2\) to \(2\). So, the symmetric point is \((-4, 2)\).

Plot the symmetric point \((-4, 2)\) on the coordinate plane. This will be 4 units to the left and 2 units above the origin.

Verify that the original point and its symmetric point are equidistant from the x-axis but on opposite sides, confirming the reflection is correct.

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주요 개념

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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 x-axis

Symmetry about the x-axis means reflecting a point across the x-axis. For a point (x, y), its symmetric point with respect to the x-axis is (x, -y). This flips the point vertically while keeping the horizontal position unchanged.

Reflection of Points

Reflection is a transformation producing a mirror image of a point across a specific line, such as the x-axis. Understanding reflection helps in visualizing and plotting symmetric points, which is essential in solving problems involving symmetry in geometry and trigonometry.