ACT Math study guide · Chapter 27
Geometric Transformations
A function transformation and a geometric transformation are easy to confuse, since both use words like shift, reflect, and stretch. A function transformation changes the graph of y = f(x) by changing the input or output of the function, as covered in the Function Transformations and Piecewise Functions chapter. A geometric transformation, the subject of this chapter, moves or resizes points and figures directly in the coordinate plane, without reference to any function rule. The vocabulary overlaps; the object being transformed does not.
A geometric transformation moves a figure according to a fixed coordinate rule, without changing its shape (except for a dilation, which changes size). Reflections, rotations, and translations preserve size and shape; only a dilation changes the figure's dimensions, scaled by a constant factor from a center point.
Each transformation has its own coordinate rule, and mixing them up is the most common error on this topic. Reflecting over the x-axis flips the sign of y; reflecting over the y-axis flips the sign of x; reflecting over the line y = x swaps the two coordinates. A 90° counterclockwise rotation about the origin sends (x, y) to (−y, x); a 180° rotation sends (x, y) to (−x, −y); a 270° counterclockwise rotation (equivalent to 90° clockwise) sends (x, y) to (y, −x).
A dilation centered at the origin with scale factor k multiplies both coordinates by k: (x, y) becomes (kx, ky). When a dilation is applied to an entire figure instead of a single point, its area scales by k², the same relationship covered in the similarity chapter of this Part.
Where this appears as a model: an image resized for printing. A 4-by-6-inch photo enlarged by a scale factor of 3 becomes 12 by 18 inches, and its area grows from 24 to 216 square inches, scaling by the square of the linear factor instead of the factor itself.
▸ Applying a rotation rule
Worked Example Point (5, −2) is rotated 90° counterclockwise about the origin. Find the image.
A 90° counterclockwise rotation follows the rule (x, y) → (−y, x).
(5, −2) → (−(−2), 5) = (2, 5)
▸ Combining two transformations
Worked Example Point (3, −4) is reflected over the y-axis, then translated 2 units down. Find the final image.
Apply each transformation in order, using the result of the first as the input to the second.
Reflect over the y-axis: (3, −4) → (−3, −4)
Translate 2 units down: (−3, −4) → (−3, −6)
Worked Example A triangle has a vertex at (0, 3). It is dilated by a scale factor of 2 centered at the origin, then reflected over the x-axis. Find the final image of that vertex.
Recognize. Two transformations are applied in sequence. Apply them one at a time, in the order given, using the result of the first as the input to the second.
Set up.
Dilate (0, 3) by scale factor 2: (0, 6)
Solve.
Reflect (0, 6) over the x-axis: (0, −6)
Check. The y-coordinate's magnitude doubled from 3 to 6, matching the dilation, and then flipped sign, matching the reflection.
Best route. Hand computation. Each rule is a one-line coordinate transformation.
Common error. Applying the two transformations in the reverse order, which does not always produce the same final image.
▸ Practice
Question 1
- AA. (−3, −5)
- BB. (3, 5)
- CC. (3, −5)
- DD. (−3, 5)
Question 2
- FF. (2, −7)
- GG. (−2, −7)
- HH. (2, 7)
- JJ. (−2, 7)
Question 3
- AA. (7, 4)
- BB. (1, −6)
- CC. (1, 4)
- DD. (9, 2)
Question 4
- FF. (6, −2)
- GG. (6, 2)
- HH. (−2, −6)
- JJ. (−6, 2)
Question 5
- AA. (15, 3)
- BB. (8, 6)
- CC. (5, 9)
- DD. (15, 9)
Question 6
- FF. (−4, −2)
- GG. (2, −4)
- HH. (4, 2)
- JJ. (−2, 4)
Question 7
- AA. (0, 4)
- BB. (0, 6)
- CC. (1, 3)
- DD. (4, 6)
Question 8
- FF. (6, −3)
- GG. (−6, −3)
- HH. (−6, 3)
- JJ. (6, 3)
Question 9
- AA. (3, 5)
- BB. (−3, 5)
- CC. (3, −5)
- DD. (−3, −5)
Question 10
- FF. 40
- GG. 50
- HH. 20
- JJ. 12.5
Question 11
- AA. (−7, −2)
- BB. (2, −7)
- CC. (−2, −7)
- DD. (−2, 7)
Question 12
- FF. (0, −6)
- GG. (−6, 0)
- HH. (0, 6)
- JJ. (6, 0)
Question 13
- AA. (1, 8)
- BB. (1, 4)
- CC. (9, −8)
- DD. (1, −4)
Question 14
- FF. (−0.5, 2)
- GG. (−1, 2)
- HH. (−2, 8)
- JJ. (−0.5, 4)
Question 15
- AA. (−3, −3)
- BB. (−3, 3)
- CC. (3, 3)
- DD. (3, −3)
Question 16
- FF. (7, 5)
- GG. (12, 6)
- HH. (12, 2)
- JJ. (4, 6)
Question 17
- AA. (5, 8)
- BB. (3, 8)
- CC. (5, 10)
- DD. (−5, −6)
Question 18
- FF. 5
- GG. 20
- HH. 10
- JJ. 2.5
Question 19
- AA. (−6, −1)
- BB. (1, −6)
- CC. (−1, 6)
- DD. (6, −1)
Question 20
- FF. (−4, 7)
- GG. (−4, −7)
- HH. (4, 7)
- JJ. (4, −7)
▸ Answers and Explanations
Show the answers and explanations
1: C. Reflecting over the x-axis keeps x the same and flips the sign of y: (3, 5) → (3, −5). Choice D reflects over the y-axis, flipping the sign of x when it should flip the sign of y.
2: H. Reflecting over the y-axis keeps y the same and flips the sign of x: (−2, 7) → (2, 7). Choice F reflects over the x-axis, flipping the sign of y when it should flip the sign of x.
3: A. Translating right adds to x; translating up adds to y: (4+3, −1+5) = (7, 4). Choice B subtracts both values instead of adding them.
4: J. A 90° counterclockwise rotation about the origin follows the rule (x, y) → (−y, x): (2, 6) → (−6, 2). Choice F applies the 90° clockwise rule, (x, y) → (y, −x), instead of the counterclockwise rule.
5: D. A dilation by scale factor k multiplies both coordinates by k: (5×3, 3×3) = (15, 9). Choice A scales only the x-coordinate, leaving y unchanged.
6: G. Reflecting over the line y = x swaps the coordinates: (−4, 2) → (2, −4). Choice F reflects over the x-axis instead of the line y = x.
7: A. Translating left subtracts from x; translating down subtracts from y: (2−2, 5−1) = (0, 4). Choice D adds both values instead of subtracting them.
8: H. A 180° rotation about the origin flips the sign of both coordinates: (6, −3) → (−6, 3). Choice G flips the sign of only the y-coordinate, leaving x unchanged.
9: B. Reflecting over the x-axis keeps x the same and flips the sign of y: (−3, −5) → (−3, 5). Choice C reflects over the y-axis, flipping the sign of x when it should flip the sign of y.
10: G. Area scales by the square of the linear scale factor: 8 × 2.5² = 8 × 6.25 = 50. Choice H applies the scale factor directly instead of squaring it, computing 8×2.5=20.
11: B. A 270° counterclockwise rotation about the origin follows the rule (x, y) → (y, −x): (7, 2) → (2, −7). Choice D applies the 90° counterclockwise rule instead of the 270° rule.
12: F. Reflecting over the y-axis flips the sign of x and keeps y the same: (0, −6) → (−0, −6) = (0, −6), unchanged since x = 0. Choice H mistakenly reflects over the x-axis instead, flipping the sign of y.
13: D. First translate: (5−4, −2+6) = (1, 4). Then reflect over the x-axis: (1, −4). Choice B completes the translation correctly but forgets to apply the reflection afterward.
14: F. A dilation by scale factor k multiplies both coordinates by k: (−1×0.5, 4×0.5) = (−0.5, 2). Choice G scales only the y-coordinate, leaving x unchanged.
15: A. Reflecting over the origin flips the sign of both coordinates: (3, 3) → (−3, −3). Choice B reflects over the y-axis only, flipping the sign of x but leaving y unchanged.
16: G. A dilation by scale factor k multiplies both coordinates by k: (4×3, 2×3) = (12, 6). Choice F adds 3 to each coordinate instead of multiplying by 3.
17: C. First rotate 90° counterclockwise: (8, −5) → (5, 8). Then translate 2 units up: (5, 10). Choice A completes the rotation correctly but forgets to apply the translation afterward.
18: F. Area scales by the square of the linear scale factor: 20 × 0.5² = 20 × 0.25 = 5. Choice H applies the scale factor directly instead of squaring it, computing 20×0.5=10.
19: B. Reflecting over the line y = x swaps the coordinates: (−6, 1) → (1, −6). Choice D flips the sign of both coordinates instead of swapping them.
20: J. First rotate 180°: (4, 7) → (−4, −7). Then reflect over the y-axis: (4, −7). Choice G completes the rotation correctly but forgets to apply the reflection afterward.