Transformations questions on the ACT come in two flavors: moving points and shapes around the coordinate plane, and shifting or stretching function graphs. This free, complete lesson covers translations, reflections, rotations, dilations, and function graph transformations. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
A translation slides a point or shape to a new position without changing its size, shape, or orientation. Add the horizontal shift to every x-coordinate and the vertical shift to every y-coordinate.
Worked Example: Point A(3, −5) is translated 4 units left and 6 units up. Find the new coordinates.
New x: 3 − 4 = −1 New y: −5 + 6 = 1
A' = (−1, 1)
1. Point B(2, 7) is translated 5 units right and 3 units down. Find the new coordinates.
Explanation: New x: 2 + 5 = 7. New y: 7 − 3 = 4. Result: (7, 4).
2. Triangle vertices A(1, 2), B(4, 2), C(4, 6) are translated 3 units left and 2 units up. Find the new coordinates of vertex C.
Explanation: New x: 4 − 3 = 1. New y: 6 + 2 = 8. Result: (1, 8).
3. A translation moves point (−2, 5) to (3, 1). What is the translation rule?
Explanation: Change in x: 3 − (−2) = 5 (right). Change in y: 1 − 5 = −4 (down).
4. Point D(0, 0) is translated using the rule (x, y) → (x + 6, y − 9). What are the new coordinates?
Explanation: (0 + 6, 0 − 9) = (6, −9).
A reflection flips a point or shape across a line, like a mirror image.
| Reflection across... | Rule |
|---|---|
| The x-axis | (a, b) → (a, −b) |
| The y-axis | (a, b) → (−a, b) |
| The line y = x | (a, b) → (b, a) |
Worked Example: Reflect point (4, −3) across the x-axis.
Reflecting across the x-axis flips the sign of the y-coordinate:
(4, −3) → (4, 3)
1. Reflect point (5, 2) across the y-axis.
Explanation: Reflecting across the y-axis flips the sign of the x-coordinate: (−5, 2).
2. Reflect point (−3, −7) across the x-axis.
Explanation: Reflecting across the x-axis flips the sign of the y-coordinate: (−3, 7).
3. Reflect point (6, −2) across the line y = x.
Explanation: Reflecting across y = x swaps the coordinates: (−2, 6).
4. A trapezoid has vertex D at (8, 3). If the trapezoid is reflected across the y-axis, what are the new coordinates of D?
Explanation: Reflecting across the y-axis flips the sign of the x-coordinate: (−8, 3).
A rotation turns a point or shape around a fixed center point, usually the origin. All ACT rotations are counterclockwise unless stated otherwise.
| Rotation about the origin | Rule |
|---|---|
| 90° counterclockwise | (a, b) → (−b, a) |
| 180° | (a, b) → (−a, −b) |
| 270° counterclockwise (= 90° clockwise) | (a, b) → (b, −a) |
Worked Example: Rotate point (5, 2) 90° counterclockwise about the origin.
(a, b) → (−b, a): (5, 2) → (−2, 5)
1. Rotate point (3, 7) 180° about the origin.
Explanation: (a, b) → (−a, −b): (3, 7) → (−3, −7).
2. Rotate point (4, −6) 90° counterclockwise about the origin.
Explanation: (a, b) → (−b, a): (4, −6) → (6, 4).
3. Rotate point (2, 9) 270° counterclockwise about the origin.
Explanation: (a, b) → (b, −a): (2, 9) → (9, −2).
4. A point is rotated 90° counterclockwise about the origin and lands at (−7, 3). What were its original coordinates?
Explanation: Working backward from (a, b) → (−b, a) = (−7, 3): −b = −7, so b = 7, and a = 3. The original point was (3, 7).
A dilation scales a point or shape by a scale factor, centered at a fixed point (usually the origin). Unlike translations, reflections, and rotations, a dilation changes size.
Worked Example: Dilate point (4, 6) by a scale factor of 3, centered at the origin.
(4 × 3, 6 × 3) = (12, 18)
1. Dilate point (5, −2) by a scale factor of 4, centered at the origin.
Explanation: (5 × 4, −2 × 4) = (20, −8).
2. Dilate point (12, −18) by a scale factor of 1/3, centered at the origin.
Explanation: (12 × 1/3, −18 × 1/3) = (4, −6).
3. A triangle is dilated by a scale factor of 2, centered at the origin. How does the new perimeter compare to the original?
Explanation: Perimeter is a linear measurement, so it scales directly with the scale factor: twice the original.
4. A figure is dilated by a scale factor of 5, centered at the origin. How does the area of the new figure compare to the original?
Explanation: Area scales by the square of the scale factor: 5² = 25 times the original.
For a function g(x) built from f(x), changes inside the parentheses affect the graph horizontally (and work opposite to what they look like); changes outside affect it vertically (and work the way they look).
| Change | Effect on the graph |
|---|---|
| f(x) + k | Shifts up k units (k > 0) |
| f(x) − k | Shifts down k units (k > 0) |
| f(x − h) | Shifts right h units (h > 0) |
| f(x + h) | Shifts left h units (h > 0) |
| −f(x) | Reflects over the x-axis |
| f(−x) | Reflects over the y-axis |
| a · f(x), |a| > 1 | Vertically stretches by a factor of a |
Worked Example: If g(x) = f(x − 4) + 2, how does the graph of g compare to f?
The graph of g is the graph of f shifted 4 units right and 2 units up.
1. If g(x) = f(x + 3) − 5, how does the graph of g compare to f?
Explanation: f(x + 3) shifts left 3 units; subtracting 5 shifts down 5 units.
2. If g(x) = −f(x), how does the graph of g compare to f?
Explanation: Negating the entire output flips the graph vertically, reflecting it over the x-axis.
3. If g(x) = f(−x), how does the graph of g compare to f?
Explanation: Negating the input flips the graph horizontally, reflecting it over the y-axis.
4. If g(x) = 3f(x), how does the graph of g compare to f?
Explanation: Multiplying the entire output by 3 stretches the graph vertically, tripling every y-value.
5. The graph of f(x) = x² is shifted 5 units right and 3 units down. What is the equation of the new function?
Explanation: Shifting right 5 replaces x with (x − 5); shifting down 3 subtracts 3 from the whole function: (x − 5)² − 3.
Translations, reflections, and rotations are rigid transformations, they preserve size and shape exactly, only changing position or orientation. A dilation is not rigid, it changes the size of the figure while preserving its overall shape.
Not typically, since a rotated function would usually fail the vertical line test (multiple y-values for one x-value) and would no longer represent a function at all. Function transformations on the ACT stick to shifts, reflections, and stretches.
The graph shifts wherever the original output now occurs. Since f(x − h) reaches the same output that f(x) reached, but h units later (at a larger x-value), the entire graph appears to move to the right.
The Transformations & Images quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.
