Describe Transformation Effects Using Coordinates - 8th Grade Math

Which transformation is represented by the rule $(x,y)\to(-x,\,y)$ ?
Answer: Reflection over the $y$ -axis 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).
Triangle $ABC$ has vertices $A(1,2)$ , $B(3,2)$ , and $C(2,4)$ . The triangle is translated by the vector $(4,3)$ to form triangle $A'B'C'$ with vertices $A'(5,5)$ , $B'(7,5)$ , and $C'(6,7)$ . What is the coordinate rule for this translation?
Answer: $(x,y)\to(x+4,\ y+3)$ 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).
A point $P(3,2)$ is rotated $90^\circ$ counterclockwise about the origin to $P'(-2,3)$ . Which rule matches this rotation?
Answer: $(x,y)\to(-y,\ x)$ 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).
Triangle $DEF$ has vertices $D(1,1)$ , $E(2,1)$ , and $F(1,3)$ . It is dilated by a scale factor of $2$ centered at the origin, forming $D'(2,2)$ , $E'(4,2)$ , and $F'(2,6)$ . What is the coordinate rule for this dilation?
Answer: $(x,y)\to(2x,\ 2y)$ 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).
A triangle has vertices $A(-1,0)$ , $B(2,1)$ , and $C(1,-3)$ . It is reflected over the line $y=x$ . What are the coordinates of $B'$ ?
Answer: $(1,2)$ 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 B(2,1) reflected over y=x: apply (x,y)→(y,x) getting B'(1,2), or over y-axis would be (-2,1). In this case, the reflection over y=x correctly applies (x,y)→(y,x) to transform B(2,1) to B'(1,2). A common error might be using the wrong reflection rule, such as (-x,y) instead of (y,x) for y=x, or swapping incorrectly to (2,-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).
A point $M(-2,3)$ is dilated by a scale factor of $2$ centered at the origin to form $M'$ . Which rule and image coordinate are correct?
Answer: $(x,y)\to(2x,\,2y)$ and $M'(-4,6)$ 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).
Square $WXYZ$ has vertices $W(1,-2)$ , $X(3,-2)$ , $Y(3,0)$ , and $Z(1,0)$ . The square is translated to $W'(-2,2)$ , $X'(0,2)$ , $Y'(0,4)$ , and $Z'(-2,4)$ . Which rule represents this translation?
Answer: $(x,y)\to(x-3,\ y+4)$ 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, square W(1,-2) translated to W'(-2,2): the change is -3 in x and +4 in y, so (x-3,y+4), applied to others confirms. In this case, the translation correctly applies the rule (x,y)→(x-3,y+4) to produce W'(-2,2), X'(0,2), Y'(0,4), and Z'(-2,4). A common error might be reversing the signs, such as (x+4,y-3) instead of (x-3,y+4), or miscalculating the vector as (x+3,y-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).
Point $P(-3, 5)$ is rotated $90°$ counterclockwise about the origin, then dilated by a scale factor of $2$ centered at the origin. If this same point had instead been dilated first by a scale factor of $2$ , then rotated $90°$ counterclockwise about the origin, what would be the relationship between the two final positions?
Answer: The two final positions would be the same point with identical coordinates Both sequences result in the same final point $(-10, -6)$ . 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.
Square $WXYZ$ has vertices $W(0, 0)$ , $X(4, 0)$ , $Y(4, 4)$ , and $Z(0, 4)$ . The square is dilated by scale factor $\frac{3}{2}$ centered at $W$ , then rotated $180°$ about the origin. What is the $x$ -coordinate of the final image of vertex $Y$ ?
Answer: $-6$ 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 $Y(4, 4)$ through both transformations. First, the dilation by scale factor $\frac{3}{2}$ centered at $W(0, 0)$ . When dilating from the origin, you multiply each coordinate by the scale factor: $Y'(4 \cdot \frac{3}{2}, 4 \cdot \frac{3}{2}) = Y'(6, 6)$ . Next, rotate this new point $180°$ about the origin. A $180°$ rotation changes $(x, y)$ to $(-x, -y)$ . So $Y'(6, 6)$ becomes $Y''(-6, -6)$ . The final $x$ -coordinate is $-6$ . Looking at the wrong answers: Choice A ( $4$ ) is the original $x$ -coordinate of $Y$ , suggesting you ignored both transformations. Choice B ( $6$ ) 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 ( $-4$ ) occurs if you mistakenly rotated the original point $Y(4, 4)$ first without dilating, giving $(-4, -4)$ . The correct answer is D ( $-6$ ). 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.
Point $M(6, -8)$ is transformed by a dilation centered at the origin with an unknown scale factor $k$ , resulting in point $M'(9, -12)$ . If this same scale factor is applied to point $N(-4, 10)$ , what are the coordinates of $N'$ ?
Answer: $(-6, 15)$ 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 $k$ , compare the original point $M(6, -8)$ with its image $M'(9, -12)$ . Since dilation multiplies each coordinate by the scale factor, you have: - For the x-coordinate: $6k = 9$ , so $k = \frac{9}{6} = \frac{3}{2}$ - For the y-coordinate: $-8k = -12$ , so $k = \frac{-12}{-8} = \frac{3}{2}$ Both coordinates give the same scale factor of $\frac{3}{2}$ , confirming our calculation. Now apply this scale factor to point $N(-4, 10)$ : - New x-coordinate: $-4 \times \frac{3}{2} = -6$ - New y-coordinate: $10 \times \frac{3}{2} = 15$ So $N'(-6, 15)$ , which is answer choice C. Looking at the wrong answers: A) $(-2, 12)$ suggests using scale factor $\frac{1}{2}$ for x and $\frac{6}{5}$ for y, mixing up the calculation. B) $(-6, 12)$ correctly finds the x-coordinate but incorrectly uses scale factor $\frac{6}{5}$ for the y-coordinate. D) $(-8, 15)$ 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.
Triangle $RST$ has vertices $R(-3,2)$ , $S(-1,5)$ , and $T(-4,4)$ . The triangle is translated by $(h,k)=(2,-3)$ . What are the coordinates of $S'$ ?
Answer: $(1,2)$ 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). Translating by (2,-3) adds to S(-1,5), resulting in S'(1,2). Errors like wrong signs in translation, such as adding -2 instead of +2, are common. To apply: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→(x+2,y-3)), (3) apply to the vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts right 2 and down 3). 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).
A student uses the rule $(x,y)\to(x-3,\ y+2)$ to move a shape on the coordinate plane. Which translation vector describes this transformation?
Answer: Translate by $(-3,2)$ 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).
Triangle $JKL$ has vertices $J(2,-1)$ , $K(4,-1)$ , and $L(3,2)$ . It is dilated by a scale factor of $2$ centered at the origin. What are the coordinates of $L'$ ?
Answer: $(6,4)$ 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).
A rectangle has vertices $(2,1)$ , $(5,1)$ , $(5,3)$ , and $(2,3)$ . It is reflected over the y-axis to form a new rectangle with vertices $(-2,1)$ , $(-5,1)$ , $(-5,3)$ , and $(-2,3)$ . Which rule represents this reflection?
Answer: $(x,y) \to (-x, y)$ This question tests describing transformation effects on coordinates using rules: translation $(x,y) \to (x+h,y+k)$ , reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by $(h,k)$ adds to coordinates $(x,y) \to (x+h,y+k)$ , reflection over y-axis negates x $(x,y) \to (-x,y)$ , over x-axis negates y $(x,y) \to (x,-y)$ , rotation 90° CCW about origin uses $(x,y) \to (-y,x)$ , dilation scale k from origin multiplies both $(x,y) \to (kx,ky)$ ; apply the rule to all vertices to get the image figure. For example, a rectangle with vertices $(2,1)$ , $(5,1)$ , $(5,3)$ , $(2,3)$ reflected over the y-axis: apply $(x,y) \to (-x,y)$ getting $(-2,1)$ , $(-5,1)$ , $(-5,3)$ , $(-2,3)$ , or translated by $(4,3)$ would be $(x+4,y+3)$ . In this case, the reflection over the y-axis correctly applies the rule $(x,y) \to (-x,y)$ to produce the image vertices $(-2,1)$ , $(-5,1)$ , $(-5,3)$ , and $(-2,3)$ . A common error might be negating the wrong coordinate, such as using $(x,-y)$ for y-axis reflection instead of $(-x,y)$ , or confusing it with $(-x,-y)$ which is a 180° rotation. To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule $((x,y) \to \dots)$ , (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 $(h,k)$ , 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).
Point $T(-3,6)$ is rotated $180^\circ$ about the origin to form $T'$ . What is the coordinate rule for this rotation?
Answer: $(x,y)\to(-x,-y)$ 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).
A triangle has vertices $A(1,2)$ , $B(3,2)$ , and $C(2,4)$ . The triangle is translated by the vector $(4,3)$ to form $A'$ , $B'$ , and $C'$ . What is the coordinate rule for this translation?
Answer: $(x,y)\to(x+4,\,y+3)$ 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 correctly maps the vertices to their images. A common error is using the wrong sign in translation, such as (x+4,y-3) instead of (x+4,y+3), or confusing it with reflection by negating coordinates. To apply the rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→(x+4,y+3)), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts the figure right 4 and up 3 units). 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).
Rectangle $PQRS$ has vertices $P(2,1)$ , $Q(5,1)$ , $R(5,3)$ , and $S(2,3)$ . The rectangle is reflected over the $y$ -axis to form $P'Q'R'S'$ . Which rule represents this reflection?
Answer: $(x,y)\to(-x,y)$ 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 reflection over the y-axis uses the rule (x,y)→(-x,y), which matches choice B. A common error is wrong coordinate negated, like for y-axis reflection negating y not x, resulting in (x,-y) instead. 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).
Quadrilateral $ABCD$ has vertices $A(-2,1)$ , $B(1,1)$ , $C(1,4)$ , and $D(-2,4)$ . It is reflected over the $x$ -axis to form $A'B'C'D'$ . What are the coordinates of $C'$ ?
Answer: $(1,-4)$ 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).
Triangle $DEF$ has vertices $D(1,1)$ , $E(2,1)$ , and $F(1,3)$ . It is dilated by a scale factor of $2$ centered at the origin to form triangle $D'E'F'$ . What are the coordinates of $F'$ ?
Answer: $(3,6)$ 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).
Point $Q(-4,5)$ is reflected over the x-axis. What are the coordinates of $Q'$ after the reflection?
Answer: $(-4,-5)$ Reflecting a point over the x-axis keeps the x-coordinate the same and flips the sign of the y-coordinate: $(x,y)\to(x,-y)$ . Applying this to $Q(-4,5)$ gives $Q'=(-4,-5)$ , so D is correct. Choice A is wrong because it flips the sign of $x$ instead of $y$ ; 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 $180°$ rotation about the origin, not a reflection over the x-axis.

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