Translate, reflect, rotate, dilate — the four moves on the coordinate plane and the rules for each. Plus the distance, midpoint, and slope formulas you need on every coordinate question.
On the coordinate plane, every “find the missing point” or “what shape is this” question becomes plug-and-chug. You only need three formulas and four transformation rules — and the Test Out tests them in the same way every year.
Three formulas to memorize
Distance: d = √[(x2 − x1)2 + (y2 − y1)2]Midpoint: M = ( (x1+x2)/2 , (y1+y2)/2 )Slope: m = (y2 − y1) / (x2 − x1)Distance is Pythagorean in disguise. Midpoint is the average. Slope is rise over run.
Parallel & perpendicular by slope
Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes (m and −1/m). Horizontal → m = 0. Vertical → m undefined.
The four transformations
Three rigid motions (translation, reflection, rotation) preserve size. Dilation scales it.
The (x, y) rules
Translate by (a, b): (x, y) → (x + a, y + b)Reflect across x-axis: (x, y) → (x, −y)Reflect across y-axis: (x, y) → (−x, y)Reflect across y = x: (x, y) → (y, x)Rotate 90° CCW about origin: (x, y) → (−y, x)Rotate 90° CW about origin: (x, y) → (y, −x)Rotate 180° about origin: (x, y) → (−x, −y)Dilate by factor k (center at origin): (x, y) → (kx, ky)
Rotation memory hook
For 90° CCW (counter-clockwise): swap, then negate the new x. CW: swap, then negate the new y. 180°: negate both. Always start by swapping x and y — the signs come last.
Translate a point
Point P(4, -2) is translated 3 units left and 5 units up. What are the coordinates of P'?
Translation: (x - 3, y + 5) = (4 - 3, -2 + 5) = (1, 3). Move left subtracts from x; move up adds to y.
Reflect a triangle
A triangle has vertices A(2, 3), B(6, 3), and C(4, 7). The triangle is reflected across the y-axis. What are the coordinates of A' (the image of A)?
Reflection across the y-axis changes (x, y) to (-x, y). So A(2, 3) becomes A'(-2, 3). The y-coordinate stays the same; only the x-coordinate sign changes.
Rotate 90° clockwise
A figure with vertices A(1, 2), B(4, 2), C(4, 5) is rotated 90° clockwise about the origin. What are the coordinates of B'?
Rotation 90° clockwise about the origin: (x, y) → (y, -x). So B(4, 2) → B'(2, -4).
Composite transformations
Order matters
If a problem says “reflect, then translate” — do them in that order. Reflecting a translated figure is not the same as translating a reflected figure. Apply each step to the result of the previous step.