Let’s solve each transformation step by step. We’ll use matrix multiplication for each one. Remember: to transform a point (x, y), we write it as a column vector [x; y] and multiply by the given matrix.
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1) Translation 7 units left and 9 units up
Translation is NOT done with a 2x2 matrix — it’s done by adding to coordinates.
Original triangle vertices (from graph):
Looking at the first graph, the triangle has vertices at:
→ (1, -3), (4, -3), (1, -1)
Translate 7 units left → subtract 7 from x
Translate 9 units up → add 9 to y
New points:
- (1 - 7, -3 + 9) = (-6, 6)
- (4 - 7, -3 + 9) = (-3, 6)
- (1 - 7, -1 + 9) = (-6, 8)
✔ Final Answer for #1: Vertices are
(-6, 6), (-3, 6), (-6, 8)
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2) Dilation of 2.5
Dilation means scaling from origin. Matrix for dilation by k is:
[ k 0 ]
[ 0 k ]
So for k = 2.5, matrix is:
[2.5 0 ]
[0 2.5]
Original triangle vertices (second graph):
Looks like: (2, 1), (4, 1), (3, 3)
Multiply each point by 2.5:
- (2×2.5, 1×2.5) = (5, 2.5)
- (4×2.5, 1×2.5) = (10, 2.5)
- (3×2.5, 3×2.5) = (7.5, 7.5)
✔ Final Answer for #2: Vertices are
(5, 2.5), (10, 2.5), (7.5, 7.5)
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3) Rotation 180° about the origin
Rotation 180° matrix is:
[-1 0]
[ 0 -1]
Original triangle vertices (third graph):
Looks like: (-3, 2), (-1, 2), (-2, 4)
Apply rotation:
- (-3 × -1, 2 × -1) = (3, -2)
- (-1 × -1, 2 × -1) = (1, -2)
- (-2 × -1, 4 × -1) = (2, -4)
✔ Final Answer for #3: Vertices are
(3, -2), (1, -2), (2, -4)
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4) Reflection across the y-axis
Reflection over y-axis matrix:
[-1 0]
[ 0 1]
Original triangle vertices (fourth graph):
Looks like: (-2, -1), (-4, -3), (-2, -4)
Apply reflection:
- (-2 × -1, -1 × 1) = (2, -1)
- (-4 × -1, -3 × 1) = (4, -3)
- (-2 × -1, -4 × 1) = (2, -4)
✔ Final Answer for #4: Vertices are
(2, -1), (4, -3), (2, -4)
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Now the bottom part: “Find the coordinates... after given transformation”
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a) Rotation 90° counterclockwise about the origin
Matrix for 90° CCW:
[ 0 -1]
[ 1 0]
Given points: (-4, 1), (-3, -2)
Apply matrix:
First point (-4, 1):
x’ = 0*(-4) + (-1)*1 = -1
y’ = 1*(-4) + 0*1 = -4
→ (-1, -4)
Second point (-3, -2):
x’ = 0*(-3) + (-1)*(-2) = 2
y’ = 1*(-3) + 0*(-2) = -3
→ (2, -3)
✔ Final Answer for a):
(-1, -4), (2, -3)
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b) Translation 3 units left and 5 units up
Again, translation = add/subtract.
Points: (2, 5), (5, 8), (6, 4)
Left 3 → subtract 3 from x
Up 5 → add 5 to y
New points:
- (2 - 3, 5 + 5) = (-1, 10)
- (5 - 3, 8 + 5) = (2, 13)
- (6 - 3, 4 + 5) = (3, 9)
✔ Final Answer for b):
(-1, 10), (2, 13), (3, 9)
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Final Answer:
1) (-6, 6), (-3, 6), (-6, 8)
2) (5, 2.5), (10, 2.5), (7.5, 7.5)
3) (3, -2), (1, -2), (2, -4)
4) (2, -1), (4, -3), (2, -4)
a) (-1, -4), (2, -3)
b) (-1, 10), (2, 13), (3, 9)
Parent Tip: Review the logic above to help your child master the concept of transformations with matrices worksheet.
