You can describe the position, shape, and size of apolygon on a coordinate plane by naming theordered pairs that define its vertices.The coordinates of ΔABC below are A (–2, –1),B (0, 3), and C (1, –2) .You can also define ΔABC by a matrix: x-coordinates y-coordinates
A translation matrix is a matrix used totranslate coordinates on the coordinate plane.The matrix sum of a preimage and a translationmatrix gives the coordinates of the translatedimage.
Reading MathThe prefix pre- means ―before,‖ so the preimageis the original figure before any transformationsare applied. The image is the resulting figureafter a transformation.
Example 1: Using Matrices to Translate a Figure Translate ΔABC with coordinates A(–2, 1), B(3, 2), and C(0, –3), 3 units left and 4 units up. Find the coordinates of the vertices of the image, and graph.The translationmatrix will have –3 x-coordinatesin all entries in row y-coordinates1 and 4 in all entriesin row 2.
Example 1 ContinuedABC, the image ofABC, has coordinatesA(–5, 5), B(0, 6), andC(–3, 1).
Check It Out! Example 1Translate ΔGHJ with coordinates G(2, 4), H(3,1), and J(1, –1) 3 units right and 1 unit down.Find the coordinates of the vertices of the imageand graph.The translationmatrix will have 3 in x-coordinatesall entries in row 1 y-coordinatesand –1 in all entriesin row 2.
Check It Out! Example 1 ContinuedGHJ, the image ofGHJ, has coordinatesG(5, 3), H(6, 0), andJ(4, –2).
A dilation is a transformation that scales—enlargesor reduces—the preimage, resulting in similarfigures. Remember that for similar figures, theshape is the same but the size may be different.Angles are congruent, and side lengths areproportional.When the center of dilation is the origin,multiplying the coordinate matrix by a scalar givesthe coordinates of the dilated image. In thislesson, all dilations assume that the origin is thecenter of dilation.
Example 2: Using Matrices to Enlarge a FigureEnlarge ΔABC with coordinatesA(2, 3), B(1, –2), and C(–3, 1), by a factorof 2. Find the coordinates of the vertices ofthe image, and graph.Multiply each coordinate by 2 by multiplying eachentry by 2. x-coordinates y-coordinates
Example 2 ContinuedABC, the image ofABC, has coordinatesA(4, 6), B(2, –4),and C(–6, 2).
Check It Out! Example 2Enlarge ΔDEF with coordinates D(2, 3), E(5,1), and F(–2, –7) a factor of . Find thecoordinates of the vertices of the image, andgraph.Multiply each coordinate by by multiplying eachentry by .
Check It Out! Example 2 ContinuedDEF, the image ofDEF, has coordinates
A reflection matrix is a matrix that creates amirror image by reflecting each vertex over aspecified line of symmetry. To reflect a figureacross the y-axis, multiplyby the coordinate matrix. This reverses the x-coordinates and keeps the y-coordinatesunchanged.
CautionMatrix multiplication is not commutative. So besure to keep the transformation matrix on theleft!
Example 3: Using Matrices to Reflect a FigureReflect ΔPQR with coordinatesP(2, 2), Q(2, –1), and R(4, 3) across they-axis. Find the coordinates of thevertices of the image, and graph. Each x-coordinate is multiplied by –1. Each y-coordinate is multiplied by 1.
Example 3 ContinuedThe coordinates of the vertices of the image areP(–2, 2), Q(–2, –1), and R(–4, 3).
Check It Out! Example 3To reflect a figure across the x-axis, multiply by .Reflect ΔJKL with coordinates J(3, 4), K(4, 2),and L(1, –2) across the x-axis. Find thecoordinates of the vertices of the image andgraph.
Check It Out! Example 3The coordinates of the vertices of the imageare J(3, –4), K(4, –2), L(1, 2).
A rotation matrix is a matrix used to rotate afigure. Example 4 gives several types of rotationmatrices.
Example 4: Using Matrices to Rotate a FigureUse each matrix to rotate polygon ABCDwith coordinates A(0, 1), B(2, –4), C(5, 1), and D(2, 3) about the origin.Graph and describe the image.A.The image ABCD is rotated 90° counterclockwise.B.The image ABCD is rotated 90° clockwise.
Lesson QuizTransform triangle PQR with verticesP(–1, –1), Q(3, 1), R(0, 3). For each, showthe matrix transformation and state thevertices of the image.1. Translation 3 units to the left and 2 units up.2. Dilation by a factor of 1.5.3. Reflection across the x-axis.4. 90° rotation, clockwise.