Mathematics
Grade 12
15 min
Transformation matrices graph the image
Transformation matrices graph the image
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Represent a polygon on a Cartesian plane using a coordinate matrix.
Identify and write the standard 2x2 matrices for reflection, rotation, and dilation.
Apply a transformation matrix to a coordinate matrix using matrix multiplication to find the coordinates of the image.
Graph the image of a polygon given its pre-image coordinates and a transformation matrix.
Compose multiple transformations by multiplying their corresponding matrices in the correct order.
Analyze the effect of the determinant of a transformation matrix on the area of the image.
Ever wonder how video game characters move and scale so smoothly on screen? 🎮 It's all powered by the elegant mathematics of transformation matrices!
This tutorial will teach you how to use matrices to perf...
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Key Concepts & Vocabulary
TermDefinitionExample
Coordinate MatrixA matrix used to represent the coordinates of a point or the vertices of a polygon. By convention, the x-coordinates are in the first row and the y-coordinates are in the second row.A triangle with vertices A(1, 2), B(5, 4), and C(3, -1) can be represented by the coordinate matrix P = \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & -1 \end{pmatrix}.
Transformation MatrixA square matrix that, when multiplied by a coordinate matrix, produces a new coordinate matrix representing the transformed figure (the image).The matrix T = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} represents a 90° counter-clockwise rotation about the origin.
Pre-imageThe original geometric figure before a transformation is applied.The square with vertices (0,0), (1...
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Core Formulas
The Transformation Equation
P' = T \cdot P
To find the image (P'), you must pre-multiply the pre-image coordinate matrix (P) by the transformation matrix (T). The order of multiplication is critical.
Rotation Matrix (Counter-clockwise)
R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}
Use this matrix to rotate a point or shape counter-clockwise by an angle θ about the origin. Ensure your calculator is in degree mode if θ is in degrees.
Dilation (Scaling) Matrix
S(k_x, k_y) = \begin{pmatrix} k_x & 0 \\ 0 & k_y \end{pmatrix}
Use this matrix to scale a shape. It multiplies all x-coordinates by a factor of k_x and all y-coordinates by a factor of k_y.
Composition of Transformations Rule
T...
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Sign Up Free to ContinueSample Practice Questions
Easy
How is a polygon with vertices P1(x1, y1), P2(x2, y2), and P3(x3, y3) conventionally represented as a coordinate matrix P?
A.3x2 matrix with rows [x1, y1], [x2, y2], [x3, y3]
B.2x3 matrix with the first row [x1, x2, x3] and the second row [y1, y2, y3]
C.2x3 matrix with the first row [y1, y2, y3] and the second row [x1, x2, x3]
D.6x1 column matrix with entries [x1, y1, x2, y2, x3, y3]
Easy
Which of the following is the standard 2x2 transformation matrix for a reflection across the x-axis?
A.\begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix}
B.\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}
C.\begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix}
D.\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}
Easy
Given a transformation matrix T and a pre-image coordinate matrix P, which equation correctly calculates the image coordinate matrix P'?
A.P' = T \cdot P
B.P' = P \cdot T
C.P' = P + T
D.P' = T^{-1} \cdot P
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