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Position and coordinates6 / 6

Where does a reflected point go?

Distinguish the changed and unchanged coordinates in reflections across the axes.

Begin with the activity ↓

Which sign changes when you change the mirror?

A vertical-axis mirror reverses the horizontal coordinate; a horizontal-axis mirror reverses the vertical coordinate. A point on the mirror stays fixed.

Imagine the vertical axis as a mirror. Point (2, 3) is two units to its right at height three. Its reflection is two units left at the same height: (-2, 3).

Across the vertical axis, horizontal position reverses and vertical position stays fixed. Reflecting (-2, 3) again returns (2, 3). Think of taking the opposite position about zero, rather than simply adding a minus sign.

Now use the horizontal axis as the mirror. Keep horizontal 2 and move to three units below it, giving (2, -3). Only the vertical coordinate reverses.

Check a point on the mirror. Reflecting (0, 3) across the vertical axis leaves it fixed because it is already on that axis. Zero is its own opposite. The origin stays fixed under either reflection.

A reflection preserves perpendicular distance from the mirror. For these axis mirrors, count grid intervals to check that distance. Reversing both coordinates is different from one reflection across one axis.

You can now read point positions and describe a simple reflection using coordinates. You may continue to the existing symmetry experiment. Before ratios and functions, revisit comparisons between quantities and multiplier rules.

Listen one paragraph at a time
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Hold on to this distinction

For a vertical mirror, height stays fixed and left/right reverses. For a horizontal mirror, horizontal position stays fixed and up/down reverses. Name the mirror first.

Change it and check

Choose A and switch the mirror between axes. Check which coordinate changes in B, and try points on the mirror too.

Coordinate grid with one-unit intervals. A (2, 3), B (−2, 3). Horizontal values increase rightward and vertical values upward.−4−4−3−3−2−2−1−1112233440HorizontalVerticalAB
Coordinate grid with one-unit intervals. A (2, 3), B (−2, 3). Horizontal values increase rightward and vertical values upward. On small screens you may scroll the diagram horizontally.
A (2, 3) → B (−2, 3). Horizontal reverses; vertical stays fixed.

The mirror is either the horizontal or vertical axis. Reflections across arbitrary slanted lines and the general distance formula are outside this lesson.

Try with paper and objects

Mark (1, 2) on grid paper and fold at the vertical axis to find its reflection. Write its coordinates, then compare with a fold at the horizontal axis.

Check and explain another example

Check your idea 1

Reflect (2, 3) across the vertical axis. What is the image?

Check your idea 2

Reflect (-1, -2) across the horizontal axis. What is the image?

Apply it in a new setting

Reflect (0, 2) across the vertical axis. What happens?

Write an answer yourself

Writing order: two numbers: horizontal, then vertical. Write horizontal, then vertical. Use a comma and - or − for a negative value.

Return to the explanation
Compare with a solution

(-3, -1) · two numbers: horizontal, then vertical

Reverse horizontal 3 to −3 and keep vertical −1.

Does your explanation include these ideas?

Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.

Continue, or pause here