9E Coordinate Problem Solving Study Guide

Key Vocabulary

Reflect — to flip a point or shape across an axis, like a mirror image. (reflejar)

Translate — to slide a point or shape to a new position without turning it. (trasladar)

Symmetry — when one half of a shape is a mirror image of the other half. (simetría)

X-axis reflection — flip across the horizontal axis. The x stays the same, but y changes sign. (reflexión sobre el eje x)

Y-axis reflection — flip across the vertical axis. The y stays the same, but x changes sign. (reflexión sobre el eje y)

Step-by-Step: Reflections

1 Reflect across the x-axis: Keep the x-value the same. Change the SIGN of the y-value.
(3, 4) → (3, −4)
2 Reflect across the y-axis: Keep the y-value the same. Change the SIGN of the x-value.
(3, 4) → (−3, 4)
3 Reflect across BOTH axes: Change the sign of BOTH x and y.
(3, 4) → (−3, −4)

Reflection Rules

Across x-axis: (x, y) → (x, −y)

Across y-axis: (x, y) → (−x, y)

Across both: (x, y) → (−x, −y)

Visual Example

             y
             |
   (−3,4)   |   (3,4)    ← original
      *      |      *
             |
  ————————+——————— x
             |
   (−3,−4)  |   (3,−4)   ← reflected across x-axis
      *      |      *
             |

Watch Out!

Reflecting across the x-axis changes y, NOT x. Think: the x-axis is horizontal, so the point flips vertically (up/down).
Reflecting across the y-axis changes x, NOT y. The y-axis is vertical, so the point flips horizontally (left/right).
A reflected point is the same distance from the axis as the original, just on the other side.
Points ON an axis don't move when reflected across that axis. (0, 5) reflected across y-axis stays (0, 5).

Practice Problems

1. Reflect (4, 7) across the x-axis. What is the new y-coordinate?

2. Reflect (−5, 3) across the y-axis. What is the new x-coordinate?

3. Point A is at (2, −6). If you reflect it across the x-axis, what is the new point? Type as (x, y).