Middle School Math Quiz: Describe Transformation Effects Using Coordinates
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Describe Transformation Effects Using CoordinatesQuestion 1 of 20

Parallelogram PQRSPQRS undergoes a transformation that maps P(3,1)P(3, 1) to P′(1,3)P'(1, 3), Q(7,1)Q(7, 1) to Q′(1,7)Q'(1, 7), and R(9,5)R(9, 5) to R′(5,9)R'(5, 9). Based on this pattern, what type of transformation was applied, and what are the coordinates of S′S'?

Rotation 90°90° counterclockwise about origin; S′(5,5)S'(5, 5)
Reflection across y=xy = x; S′(5,5)S'(5, 5)
Reflection across y=xy = x; S′(1,5)S'(1, 5)
Translation followed by reflection; S′(1,5)S'(1, 5)
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Middle School Math Quiz

Middle School Math Quiz: Describe Transformation Effects Using Coordinates

Practice Describe Transformation Effects Using Coordinates in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Describe Transformation Effects Using Coordinates, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Parallelogram PQRSPQRS undergoes a transformation that maps P(3,1)P(3, 1) to P′(1,3)P'(1, 3), Q(7,1)Q(7, 1) to Q′(1,7)Q'(1, 7), and R(9,5)R(9, 5) to R′(5,9)R'(5, 9). Based on this pattern, what type of transformation was applied, and what are the coordinates of S′S'?

  1. Rotation 90°90° counterclockwise about origin; S′(5,5)S'(5, 5)
  2. Reflection across y=xy = x; S′(5,5)S'(5, 5) (correct answer)
  3. Reflection across y=xy = x; S′(1,5)S'(1, 5)
  4. Translation followed by reflection; S′(1,5)S'(1, 5)
Explanation: When you encounter coordinate transformation problems, your first step is to analyze how each point changes to identify the pattern. Look at what happens to the x and y coordinates systematically. Let's examine the given transformations: P(3,1)→P′(1,3)P(3,1) \to P'(1,3), Q(7,1)→Q′(1,7)Q(7,1) \to Q'(1,7), and R(9,5)→R′(5,9)R(9,5) \to R'(5,9). Notice that in each case, the x-coordinate and y-coordinate are swapped. This is the defining characteristic of a reflection across the line y=xy = x. To find S′S', we first need the coordinates of point SS. Since PQRSPQRS is a parallelogram, opposite sides are parallel and equal. Using the given points, we can determine that S(5,5)S(5,5). When reflected across y=xy = x, point S(5,5)S(5,5) becomes S′(5,5)S'(5,5) because swapping coordinates that are equal gives the same point. Now let's examine why the other answers are incorrect. Choice A suggests a 90°90° counterclockwise rotation about the origin, which would map (x,y)(x,y) to (−y,x)(-y,x). This would send P(3,1)P(3,1) to P′(−1,3)P'(-1,3), not (1,3)(1,3). Choice C has the correct transformation but gives S′(1,5)S'(1,5) instead of (5,5)(5,5) – this reflects a calculation error in finding point SS. Choice D mentions a translation followed by reflection, but the pattern shows a simple single transformation. Study tip: When identifying transformations, always check what happens to coordinates systematically. Reflection across y=xy = x always swaps x and y coordinates, making it easy to spot this transformation type.

Question 2

Triangle ABCABC has vertices A(4,−1)A(4,-1), B(6,−1)B(6,-1), and C(5,2)C(5,2). It is reflected over the yy-axis to form triangle A′B′C′A'B'C'. What are the coordinates of C′C'?

  1. (−5,2)(-5,2) (correct answer)
  2. (5,−2)(5,-2)
  3. (−2,5)(-2,5)
  4. (2,−5)(2,-5)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, reflecting C(5,2) over the y-axis gives C'(-5,2) using (x,y)→(-x,y), which matches choice B. A common error is wrong coordinate negated, like negating y instead of x for y-axis reflection, resulting in (5,-2). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

Question 3

Point Q(−4,5)Q(-4,5) is reflected over the x-axis. What are the coordinates of Q′Q' after the reflection?

  1. (4,5)(4,5)
  2. (−4,5)(-4,5)
  3. (4,−5)(4,-5)
  4. (−4,−5)(-4,-5) (correct answer)
Explanation: Reflecting a point over the x-axis keeps the x-coordinate the same and flips the sign of the y-coordinate: (x,y)→(x,−y)(x,y)\to(x,-y). Applying this to Q(−4,5)Q(-4,5) gives Q′=(−4,−5)Q'=(-4,-5), so D is correct. Choice A is wrong because it flips the sign of xx instead of yy; that's what reflecting over the y-axis would do. Choice B is wrong because neither coordinate changed, so no reflection actually occurred. Choice C is wrong because it flips the sign of both coordinates, which describes a 180°180° rotation about the origin, not a reflection over the x-axis.

Question 4

A rectangle has vertices (2,1)(2,1), (5,1)(5,1), (5,3)(5,3), and (2,3)(2,3). It is reflected over the y-axis to form a new rectangle with vertices (−2,1)(-2,1), (−5,1)(-5,1), (−5,3)(-5,3), and (−2,3)(-2,3). Which rule represents this reflection?

  1. (x,y)→(y,x)(x,y) \to (y, x)
  2. (x,y)→(−x,y)(x,y) \to (-x, y) (correct answer)
  3. (x,y)→(x,−y)(x,y) \to (x, -y)
  4. (x,y)→(−x,−y)(x,y) \to (-x, -y)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k)(x,y) \to (x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k)(h,k) adds to coordinates (x,y)→(x+h,y+k)(x,y) \to (x+h,y+k), reflection over y-axis negates x (x,y)→(−x,y)(x,y) \to (-x,y), over x-axis negates y (x,y)→(x,−y)(x,y) \to (x,-y), rotation 90° CCW about origin uses (x,y)→(−y,x)(x,y) \to (-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky)(x,y) \to (kx,ky); apply the rule to all vertices to get the image figure. For example, a rectangle with vertices (2,1)(2,1), (5,1)(5,1), (5,3)(5,3), (2,3)(2,3) reflected over the y-axis: apply (x,y)→(−x,y)(x,y) \to (-x,y) getting (−2,1)(-2,1), (−5,1)(-5,1), (−5,3)(-5,3), (−2,3)(-2,3), or translated by (4,3)(4,3) would be (x+4,y+3)(x+4,y+3). In this case, the reflection over the y-axis correctly applies the rule (x,y)→(−x,y)(x,y) \to (-x,y) to produce the image vertices (−2,1)(-2,1), (−5,1)(-5,1), (−5,3)(-5,3), and (−2,3)(-2,3). A common error might be negating the wrong coordinate, such as using (x,−y)(x,-y) for y-axis reflection instead of (−x,y)(-x,y), or confusing it with (−x,−y)(-x,-y) which is a 180° rotation. To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→… )((x,y) \to \dots), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k)(h,k), x-axis reflection (x,−y)(x,-y), y-axis reflection (−x,y)(-x,y), 90° CCW rotation (−y,x)(-y,x), 180° rotation (−x,−y)(-x,-y), dilation scale k is (kx,ky)(kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (−y,x)(-y,x) not (y,−x)(y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 5

A point P(3,2)P(3,2) is rotated 90∘90^\circ counterclockwise about the origin to P′(−2,3)P'(-2,3). Which rule matches this rotation?

  1. (x,y)→(−x, −y)(x,y)\to(-x,\ -y)
  2. (x,y)→(y, −x)(x,y)\to(y,\ -x)
  3. (x,y)→(x, −y)(x,y)\to(x,\ -y)
  4. (x,y)→(−y, x)(x,y)\to(-y,\ x) (correct answer)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, point P(3,2) rotated 90° CCW: apply (x,y)→(-y,x) getting P'(-2,3), or reflected over x-axis would be (3,-2). In this case, the 90° CCW rotation correctly applies the rule (x,y)→(-y,x) to transform P(3,2) to P'(-2,3). A common error might be using the wrong rotation formula, such as (y,-x) instead of (-y,x) for 90° CCW, or confusing it with (-x,-y) for 180°. To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 6

Triangle DEFDEF has vertices D(1,1)D(1,1), E(2,1)E(2,1), and F(1,3)F(1,3). It is dilated by a scale factor of 22 centered at the origin, forming D′(2,2)D'(2,2), E′(4,2)E'(4,2), and F′(2,6)F'(2,6). What is the coordinate rule for this dilation?

  1. (x,y)→(2x, y)(x,y)\to(2x,\ y)
  2. (x,y)→(2x, 2y)(x,y)\to(2x,\ 2y) (correct answer)
  3. (x,y)→(x+2, y+2)(x,y)\to(x+2,\ y+2)
  4. (x,y)→(x, 2y)(x,y)\to(x,\ 2y)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, triangle D(1,1), E(2,1), F(1,3) dilated by scale factor 2: apply (x,y)→(2x,2y) getting D'(2,2), E'(4,2), F'(2,6), or translated by (4,3) would add to each. In this case, the dilation by scale factor 2 correctly applies the rule (x,y)→(2x,2y) to produce D'(2,2), E'(4,2), and F'(2,6). A common error might be confusing dilation with addition, such as writing (x+2,y+2) instead of (2x,2y), or applying it to only one coordinate like (2x,y). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 7

Triangle ABCABC has vertices A(1,2)A(1,2), B(3,2)B(3,2), and C(2,4)C(2,4). The triangle is translated by the vector (4,3)(4,3) to form triangle A′B′C′A'B'C' with vertices A′(5,5)A'(5,5), B′(7,5)B'(7,5), and C′(6,7)C'(6,7). What is the coordinate rule for this translation?

  1. (x,y)→(x+3, y+4)(x,y)\to(x+3,\ y+4)
  2. (x,y)→(x−4, y−3)(x,y)\to(x-4,\ y-3)
  3. (x,y)→(x+4, y−3)(x,y)\to(x+4,\ y-3)
  4. (x,y)→(x+4, y+3)(x,y)\to(x+4,\ y+3) (correct answer)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, triangle A(1,2), B(3,2), C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5), B'(7,5), C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2), B'(-3,2), C'(-2,4). In this case, the translation by (4,3) correctly applies the rule (x,y)→(x+4,y+3) to produce the image vertices A'(5,5), B'(7,5), and C'(6,7). A common error might be using the wrong sign in translation, such as (x+3,y+4) instead of (x+4,y+3), or confusing it with (x+4,y-3) by flipping the y-direction. To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 8

A student uses the rule (x,y)→(x−3, y+2)(x,y)\to(x-3,\ y+2) to move a shape on the coordinate plane. Which translation vector describes this transformation?

  1. Translate by (2,−3)(2,-3)
  2. Translate by (−2,3)(-2,3)
  3. Translate by (3,2)(3,2)
  4. Translate by (−3,2)(-3,2) (correct answer)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, the rule (x,y)→(x-3,y+2) corresponds to translation by (-3,2), or (x+4,y+3) would be (4,3). In this case, the rule (x,y)→(x-3,y+2) correctly represents translation by the vector (-3,2). A common error might be misreading the signs, such as thinking (x-3,y+2) is (3,2) instead of (-3,2), or confusing with (2,-3). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 9

Quadrilateral ABCDABCD has vertices A(−2,1)A(-2,1), B(1,1)B(1,1), C(1,4)C(1,4), and D(−2,4)D(-2,4). It is reflected over the xx-axis to form A′B′C′D′A'B'C'D'. What are the coordinates of C′C'?

  1. (1,4)(1,4)
  2. (4,1)(4,1)
  3. (1,−4)(1,-4) (correct answer)
  4. (−1,4)(-1,4)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). Reflecting over the x-axis applies (x,y)→(x,-y), so C(1,4) becomes C'(1,-4). A common error is negating x instead of y for x-axis reflection. Applying the rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→(x,-y)), (3) apply to the vertex (substitute coordinates, calculate image), (4) verify reasonable (reflection flips over x-axis). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky).

Question 10

Square WXYZWXYZ has vertices W(0,0)W(0, 0), X(4,0)X(4, 0), Y(4,4)Y(4, 4), and Z(0,4)Z(0, 4). The square is dilated by scale factor 32\frac{3}{2} centered at WW, then rotated 180°180° about the origin. What is the xx-coordinate of the final image of vertex YY?

  1. 44
  2. 66
  3. −4-4
  4. −6-6 (correct answer)
Explanation: When you encounter transformation problems involving multiple steps, you need to apply each transformation in order and track how the coordinates change at each step. Let's trace vertex Y(4,4)Y(4, 4) through both transformations. First, the dilation by scale factor 32\frac{3}{2} centered at W(0,0)W(0, 0). When dilating from the origin, you multiply each coordinate by the scale factor: Y′(4⋅32,4⋅32)=Y′(6,6)Y'(4 \cdot \frac{3}{2}, 4 \cdot \frac{3}{2}) = Y'(6, 6). Next, rotate this new point 180°180° about the origin. A 180°180° rotation changes (x,y)(x, y) to (−x,−y)(-x, -y). So Y′(6,6)Y'(6, 6) becomes Y′′(−6,−6)Y''(-6, -6). The final xx-coordinate is −6-6. Looking at the wrong answers: Choice A (44) is the original xx-coordinate of YY, suggesting you ignored both transformations. Choice B (66) is what you'd get if you only applied the dilation but forgot the rotation—a common error when working through multi-step transformations. Choice C (−4-4) occurs if you mistakenly rotated the original point Y(4,4)Y(4, 4) first without dilating, giving (−4,−4)(-4, -4). The correct answer is D (−6-6). Study tip: For multi-step transformations, always work systematically through each step and write down the intermediate coordinates. Don't try to combine steps mentally—it's easy to lose track of which transformation affects which coordinates, especially when mixing dilations and rotations.

Question 11

Point P(−3,5)P(-3, 5) is rotated 90°90° counterclockwise about the origin, then dilated by a scale factor of 22 centered at the origin. If this same point had instead been dilated first by a scale factor of 22, then rotated 90°90° counterclockwise about the origin, what would be the relationship between the two final positions?

  1. The two final positions would be the same point with identical coordinates (correct answer)
  2. The two final positions would be different points that are reflections of each other
  3. The two final positions would be different points at different distances from the origin
  4. The two final positions would be different points but at the same distance from the origin
Explanation: Both sequences result in the same final point (−10,−6)(-10, -6). Rotation and dilation about the origin commute because both preserve ratios of distances from the origin. Choice B suggests reflection symmetry that doesn't exist. Choice C incorrectly suggests different distances from origin. Choice D is partially correct about equal distances but wrong about the positions being different.

Question 12

Point M(6,−8)M(6, -8) is transformed by a dilation centered at the origin with an unknown scale factor kk, resulting in point M′(9,−12)M'(9, -12). If this same scale factor is applied to point N(−4,10)N(-4, 10), what are the coordinates of N′N'?

  1. (−2,12)(-2, 12)
  2. (−6,12)(-6, 12)
  3. (−6,15)(-6, 15) (correct answer)
  4. (−8,15)(-8, 15)
Explanation: When you see a dilation problem, you're working with transformations that resize figures while keeping them centered at a specific point. The key is finding the scale factor by comparing corresponding coordinates. To find the scale factor kk, compare the original point M(6,−8)M(6, -8) with its image M′(9,−12)M'(9, -12). Since dilation multiplies each coordinate by the scale factor, you have:
  • For the x-coordinate: 6k=96k = 9, so k=96=32k = \frac{9}{6} = \frac{3}{2}
  • For the y-coordinate: −8k=−12-8k = -12, so k=−12−8=32k = \frac{-12}{-8} = \frac{3}{2}
Both coordinates give the same scale factor of 32\frac{3}{2}, confirming our calculation. Now apply this scale factor to point N(−4,10)N(-4, 10):
  • New x-coordinate: −4×32=−6-4 \times \frac{3}{2} = -6
  • New y-coordinate: 10×32=1510 \times \frac{3}{2} = 15
So N′(−6,15)N'(-6, 15), which is answer choice C. Looking at the wrong answers: A) (−2,12)(-2, 12) suggests using scale factor 12\frac{1}{2} for x and 65\frac{6}{5} for y, mixing up the calculation. B) (−6,12)(-6, 12) correctly finds the x-coordinate but incorrectly uses scale factor 65\frac{6}{5} for the y-coordinate. D) (−8,15)(-8, 15) gets the y-coordinate right but doubles the x-coordinate instead of applying the scale factor. Remember: in dilation problems, always verify your scale factor works for both coordinates of the given transformation before applying it to the new point. This prevents calculation errors and ensures consistency.

Question 13

Triangle ABCABC has vertices A(1,2)A(1,2), B(3,2)B(3,2), and C(2,4)C(2,4). The triangle is translated by the vector (4,3)(4,3) to form triangle A′B′C′A'B'C'. What is the coordinate rule for this translation?

  1. (x,y)→(x+4, y+3)(x,y)\to(x+4,\,y+3) (correct answer)
  2. (x,y)→(x+3, y+4)(x,y)\to(x+3,\,y+4)
  3. (x,y)→(x−3, y+4)(x,y)\to(x-3,\,y+4)
  4. (x,y)→(x−4, y−3)(x,y)\to(x-4,\,y-3)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, the translation by (4,3) uses the rule (x,y)→(x+4,y+3), which matches choice C. A common error is sign wrong in translation, like writing (x+4,y+3) as (x-4,y-3) or mixing the values like (x+3,y+4). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

Question 14

Point T(−3,6)T(-3,6) is rotated 180∘180^\circ about the origin to form T′T'. What is the coordinate rule for this rotation?

  1. (x,y)→(x,−y)(x,y)\to(x,-y)
  2. (x,y)→(−x,−y)(x,y)\to(-x,-y) (correct answer)
  3. (x,y)→(−y,x)(x,y)\to(-y,x)
  4. (x,y)→(y,−x)(x,y)\to(y,-x)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, the 180° rotation uses the rule (x,y)→(-x,-y), which matches choice B. A common error is rotation formula wrong, like confusing 180° with 90° as (-y,x) instead of (-x,-y). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

Question 15

A point R(6,−1)R(6,-1) is rotated 180∘180^\circ about the origin. What are the coordinates of R′R' after the rotation?

  1. (1,-6)
  2. (6,1)
  3. (-6,1) (correct answer)
  4. (-1,6)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, point R(6,-1) rotated 180°: apply (x,y)→(-x,-y) getting R'(-6,1), or 90° CCW would be (1,6). In this case, the 180° rotation correctly applies (x,y)→(-x,-y) to transform R(6,-1) to R'(-6,1). A common error might be using the wrong rotation formula, such as (-y,x) for 180° instead of (-x,-y), or miscalculating signs to (6,1). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 16

A point M(−2,3)M(-2,3) is dilated by a scale factor of 22 centered at the origin to form M′M'. Which rule and image coordinate are correct?

  1. (x,y)→(−2x, 2y)(x,y)\to(-2x,\,2y) and M′(4,6)M'(4,6)
  2. (x,y)→(2x, 2y)(x,y)\to(2x,\,2y) and M′(−4,6)M'(-4,6) (correct answer)
  3. (x,y)→(2x, y)(x,y)\to(2x,\,y) and M′(−4,3)M'(-4,3)
  4. (x,y)→(x+2, y+2)(x,y)\to(x+2,\,y+2) and M′(0,5)M'(0,5)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). The correct rule for dilation by 2 is (x,y)→(2x,2y), transforming M(-2,3) to M'(-4,6). A mistake could be using addition like (x+2,y+2) or negating incorrectly in other rules. Applying the rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→(2x,2y)), (3) apply to the point (substitute coordinates, calculate image), (4) verify reasonable (dilation scales by 2 from origin). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky).

Question 17

Triangle DEFDEF has vertices D(1,1)D(1,1), E(2,1)E(2,1), and F(1,3)F(1,3). It is dilated by a scale factor of 22 centered at the origin to form triangle D′E′F′D'E'F'. What are the coordinates of F′F'?

  1. (3,6)(3,6) (correct answer)
  2. (1,6)(1,6)
  3. (2,6)(2,6)
  4. (2,5)(2,5)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, dilating F(1,3) by scale factor 2 gives F'(2,6) using (x,y)→(2x,2y), which matches choice A. A common error is dilation as addition, like writing (2x,2y) as (x+2,y+2). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

Question 18

Triangle JKLJKL has vertices J(2,−1)J(2,-1), K(4,−1)K(4,-1), and L(3,2)L(3,2). It is dilated by a scale factor of 22 centered at the origin. What are the coordinates of L′L'?

  1. (3,4)(3,4)
  2. (6,2)(6,2)
  3. (5,4)(5,4)
  4. (6,4)(6,4) (correct answer)
Explanation: This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, point L(3,2) dilated by scale factor 2: apply (x,y)→(2x,2y) getting L'(6,4), or by 3 would be (9,6). In this case, the dilation by scale factor 2 correctly applies (x,y)→(2x,2y) to transform L(3,2) to L'(6,4). A common error might be applying the scale to only one coordinate, such as (6,2) or (3,4), or adding instead like (5,4). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Question 19

Which transformation is represented by the rule (x,y)→(−x, y)(x,y)\to(-x,\,y)?

  1. Reflection over the xx-axis
  2. Translation right 1 unit and up 1 unit
  3. Rotation 90∘90^\circ counterclockwise about the origin
  4. Reflection over the yy-axis (correct answer)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). The rule (x,y)→(-x,y) corresponds to reflection over the y-axis, as it negates only the x-coordinate. A common error is confusing it with x-axis reflection, which negates y, or with rotation formulas. Applying the rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→(-x,y)), (3) apply to points (substitute coordinates, calculate image), (4) verify reasonable (reflection flips over y-axis). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky).

Question 20

Point M(3,2)M(3,2) is rotated 90∘90^\circ counterclockwise about the origin to form M′M'. What are the coordinates of M′M'?

  1. (2,−3)(2,-3)
  2. (−3,−2)(-3,-2)
  3. (−2,3)(-2,3) (correct answer)
  4. (3,−2)(3,-2)
Explanation: Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, rotating M(3,2) 90° CCW gives M'(-2,3) using (x,y)→(-y,x), which matches choice B. A common error is rotation formula wrong, like using 90° as (y,-x) not (-y,x). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).