Study guide
10.3 Dilations, Rotations, Translations, and Reflections
ACCUPLACER Advanced Algebra and Functions Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Scoring, Skills Insight Bands, and Placement 1.3 How to Study for AAF
2 Linear Equations
2.1 Writing Linear Equations in One and Two Variables 2.2 Solving Linear Equations 2.3 Linear Inequalities 2.4 Systems of Two Linear Equations
3 Linear Applications and Graphs
3.1 Slope, Intercepts, and Graphing Lines 3.2 Parallel and Perpendicular Lines 3.3 Modeling with Linear Functions 3.4 Graphing Linear Inequalities and Systems
4 Factoring
4.1 GCF, Grouping, and Special Products 4.2 Factoring Quadratic Trinomials 4.3 Factoring Cubics and Higher Polynomials
5 Quadratics
5.1 Quadratic Forms, Graphs, Vertex, and Intercepts 5.2 Solving Quadratics by Factoring 5.3 Quadratic Formula and Completing the Square 5.4 Quadratic Inequalities and Quadratic-Linear Systems
6 Functions
6.1 Function Notation, Evaluation, Domain, and Range 6.2 Graphs of Functions, Vertical Line Test, Maxima and Minima 6.3 Function Operations, Composition, and Inverses 6.4 Transformations and Interpreting Functions in Context
7 Radical and Rational Equations
7.1 Simplifying Radical Expressions 7.2 Solving Radical Equations and Extraneous Roots 7.3 Rational Expressions and Equations 7.4 Graphs, Domain, and Range of Radical and Rational Functions
8 Polynomial Equations
8.1 Polynomial Operations and Structure 8.2 Solving Polynomial Equations 8.3 Graphing Polynomials, Roots, and End Behavior
9 Exponential and Logarithmic Equations
9.1 Exponential Functions, Growth, and Decay 9.2 Graphs of Exponential Functions 9.3 Logarithms and Logarithmic Properties 9.4 Solving Exponential and Logarithmic Equations
10 Geometry Concepts for Algebra 1
10.1 Area, Perimeter, and Prism Volume 10.2 Distance Formula and Pythagorean Theorem 10.3 Dilations, Rotations, Translations, and Reflections
11 Geometry Concepts for Algebra 2
11.1 Volume of Cylinders, Cones, and Spheres 11.2 Intersecting Lines and Angle Relationships 11.3 Triangle Similarity and Congruence 11.4 Circles in the Coordinate Plane
12 Trigonometry
12.1 Right-Triangle Trigonometry and Special Triangles 12.2 Unit Circle, Radians, and Arc Length 12.3 Trigonometric Graphs and Identities 12.4 Solving Trig Equations; Law of Sines and Cosines
10.3 Dilations, Rotations, Translations, and Reflections
Key Takeaways Explain This Topic AI
A 90° counterclockwise rotation about the origin sends (x, y) to (−y, x); (2, −3) maps to (3, 2).
Translation by (h, k) is (x + h, y + k); reflection over the x-axis is (x, −y), over the y-axis is (−x, y), and over y = x is (y, x).
Rotation 180° about the origin is (−x, −y); 270° counterclockwise (equivalently 90° clockwise) is (y, −x).
Dilation about the origin by k is (kx, ky); |k| > 1 enlarges, 0 < |k| < 1 shrinks, and k < 0 includes a 180° turn.
These rules move geometric figures in the plane; function maps such as y = f(x) + k are the same coordinate idea applied to a graph, taught in the functions chapter, not a replacement for this table.
Last updated: August 2026
10.3 Dilations, Rotations, Translations, and Reflections
The third Algebra 1 geometry skill on College Board Table 11 is evaluating dilations, rotations, translations, and reflections . On AAF this is still part of the 5–10% / 1–2 item Geometry-for-Algebra-1 slice, so a CAT item may hand you a single point, a polygon’s vertices, or a short mapping rule and ask for the image.
This section is geometric . Points live in the coordinate plane. Function transformations such as y = f(x) + k or y = f(x − h) use the same coordinate arithmetic on a graph; that algebra is taught in Transformations and Interpreting Functions in Context . Use it only as an analogy here. If the stem says “rotate the point” or “reflect the triangle,” apply the table below, not a rewritten f .
Work FREE mixed AAF items at /practice/accuplacer-advanced-algebra . Circles written as equations in the plane, and nonprism volume, continue in Algebra 2 geometry at Volume of Cylinders, Cones, and Spheres and Circles in the Coordinate Plane .
Coordinate rules (memorize this table)
Unless the stem names a different center, AAF rotations and dilations in this skill are about the origin .
- Transformation Image of (x, y) What it does
- Translation by (h, k) (x + h, y + k) Slides; every point moves the same vector
- Reflection over the x-axis (x, −y) Vertical flip; x stays
- Reflection over the y-axis (−x, y) Horizontal flip; y stays
- Reflection over the line y = x (y, x) Swaps coordinates
- Reflection over the line y = −x (−y, −x) Swap, then change both signs
- Rotation 90° counterclockwise about the origin (−y, x) Left quarter-turn
- Rotation 180° about the origin (−x, −y) Half-turn; same as dilation by k = −1
- Rotation 270° counterclockwise about the origin (y, −x) Same as 90° clockwise
- Dilation about the origin by k (kx, ky) Scale from (0, 0)
Rigid motions (translations, rotations, reflections) preserve distance and angle measure. A dilation with |k| ≠ 1 does not: lengths multiply by |k| , areas multiply by k² . Orientation (clockwise versus counterclockwise labeling of a triangle) is preserved by translations, rotations, and dilations with k > 0 ; a reflection reverses orientation.
Translations: (x + h, y + k)
A translation adds a horizontal h and a vertical k to every point. Positive h moves right; negative h moves left. Positive k moves up; negative k moves down.
Translate (4, −1) by (−3, 5) :
(4 − 3, −1 + 5) = (1, 4) .
The vector is the same for every vertex of a polygon. If triangle ABC has vertices (0, 0) , (6, 0) , and (0, 4) , a translation by (2, −3) produces (2, −3) , (8, −3) , and (2, 1) . Distances among vertices do not change: the legs are still 6 and 4, the hypotenuse is still √(36 + 16) = √52 .
Analogy, then stop: translating a function graph up by 5 is applying (x, y) → (x, y + 5) to every point (x, f(x)) , which is y = f(x) + 5 . On a geometry item you are not rewriting a formula; you are adding h and k to listed coordinates.
Reflections
Over the x-axis: keep x , negate y . (5, −2) maps to (5, 2) . Points on the x-axis stay put.
Over the y-axis: negate x , keep y . (5, −2) maps to (−5, −2) . Points on the y-axis stay put.
Over y = x: swap. (5, −2) maps to (−2, 5) . Points already on y = x stay put. This is the geometric twin of inverse-function graphs, but you do not need f⁻¹ here.
Over y = −x: (x, y) → (−y, −x) . (5, −2) maps to (2, −5) .
Worked set from one seed point (2, −3) :
- x-axis: (2, 3)
- y-axis: (−2, −3)
- y = x: (−3, 2)
- y = −x: (3, −2)
Those four images are easy to mix up. Write the rule first, then substitute. “Flip over y-axis” is not “swap coordinates,” and “swap” is not “negate both.”
Rotations about the origin
90° counterclockwise: (x, y) → (−y, x) .
Worked example: rotate (2, −3) 90° counterclockwise about the origin.
x = 2 , y = −3 , so −y = −(−3) = 3 and the second coordinate is x = 2 .
Image: (3, 2) .
A 90° clockwise rotation is the opposite rule (x, y) → (y, −x) , which is also 270° counterclockwise. For (2, −3) that image is (−3, −2) , not (3, 2) . Clockwise versus counterclockwise is the highest-yield rotation trap on AAF. Unless the stem says clockwise, 90° means counterclockwise in standard position.
180°: (x, y) → (−x, −y) . (2, −3) maps to (−2, 3) . Two 90° counterclockwise turns: (2, −3) → (3, 2) → (−2, 3) .
270° counterclockwise: (x, y) → (y, −x) . (2, −3) maps to (−3, −2) . Three 90° turns land there; one 90° clockwise turn is shorter.
Check with a point on the positive x-axis: (4, 0) rotated 90° counterclockwise should land on the positive y-axis at (0, 4) . The rule (−0, 4) = (0, 4) confirms. If your rule sent (4, 0) to (0, −4) , you rotated clockwise.
Dilations about the origin
A dilation about the origin by scale factor k multiplies both coordinates: (x, y) → (kx, ky) .
|k| > 1 enlarges.
0 < |k| < 1 shrinks.
k = 1 is the identity.
k = 0 collapses everything to the origin.
k < 0 scales by |k| and rotates 180° (because both signs flip).
Dilate (3, −4) by k = 1/2 : (3/2, −2) . Dilate the same point by k = 3 : (9, −12) . Dilate by k = −2 : (−6, 8) .
Triangle (0, 0) , (4, 0) , (0, 6) dilated by k = 1/2 becomes (0, 0) , (2, 0) , (0, 3) . Each length is halved; the area (1/2)(4)(6) = 12 becomes (1/2)(2)(3) = 3 , which is 12 · (1/2)² . Area scale factor k² is the reason a k = 2 dilation multiplies area by 4, not by 2.
The origin is a fixed point of every dilation about the origin. If a vertex is already (0, 0) , it does not move. Dilations about a different center are possible in a full geometry course; AAF Algebra 1 items in this skill use the origin unless the stem is explicit.
Sequences
Apply maps from the inside out , the same way you compose functions, but on points.
Example: start at (2, −3) , rotate 90° counterclockwise, then translate by (1, 4) .
Rotate: (2, −3) → (3, 2) .
Translate: (3 + 1, 2 + 4) = (4, 6) .
Reversing the order changes the answer. Translate first: (2, −3) → (3, 1) , then rotate 90° counterclockwise: (−1, 3) . Composition of geometric maps is not commutative in general. If options differ by order, reconstruct the sequence in the order written.
Function-transformation analogy (do not switch chapters)
- Geometric rule Function-graph cousin
- (x, y) → (x, y + k) y = f(x) + k
- (x, y) → (x + h, y) y = f(x − h) shifts the graph right by h
- (x, y) → (x, −y) y = −f(x)
- (x, y) → (−x, y) y = f(−x)
- (x, y) → (kx, ky) with k > 0 a homothety of the graph, not a standard AAF function-shift
Those cousins help you remember signs. They do not let you skip the point-mapping table. A stem that lists vertices of quadrilateral WXYZ wants image vertices, not a new equation y = −f(x) .
Traps
90° clockwise instead of counterclockwise. (2, −3) clockwise is (−3, −2) , not (3, 2) .
Negating the wrong coordinate on a reflection. x-axis changes y ; y-axis changes x .
Swapping when you should negate , or negating when you should swap.
Dilating only one coordinate , or adding k instead of multiplying.
Using a non-origin center when the stem said “about the origin.”
Treating a dilation as a translation. (x, y) → (x + 2, y + 2) is not a dilation by 2.
College Board’s calculator policy is almost irrelevant here: these maps are integer arithmetic. Handheld calculators are not allowed except with an approved accommodation.
When you are ready for volume of cylinders, cones, and spheres, continue at nonprism-volume . Mixed CAT practice: /practice/accuplacer-advanced-algebra .