Download Transformation rules and matrices

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June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 1
Transformation rules and matrices
Objective: Write algebraic rules representing transformations, and express those rules using
matrix multiplication.
Transformation rules
Previously we have studied transformations in the xy-plane, such as translations, reflections,
rotations, and dilations. In Chapter 1, we learned how to modify a function formula in order to
transform the function’s graph. But what we haven’t addressed yet is how to give an algebraic
description of the transformation itself.
A transformation rule is a description of a transformation that tells how to get the output point
for any given input point. The following notation is typically used:
(x, y) stands for the input point.
(x′, y′) stands for the output point, also called the image.
One way to give a transformation rule is to write a pair of equations. Here is an example:
x′ = x + 3
y′ = y – 1
These equations say that given any point (x, y), the image point (x′, y′) is found by adding 3 to
the x-coordinate and subtracting 1 from the y-coordinate. (How would you describe this
transformation geometrically?)
Many geometrically interesting transformations have fairly simple equations. In fact, all
translations, reflections, rotations, and dilations can be described using equations of the form
x′ = ax + by + h
y′ = cx + dy + k
An important fact about transformation equations having this form is that the image of a
line segment is always a line segment. Therefore, if you want to see the effect of a
transformation on a polygon, you just need to find the images of the vertices.
Problems
1. For each transformation rule below, geometrically describe the transformation.
It may help to try some examples of input-output calculations.
a. x′ = –x
y′ = y
b. x′ = 2x
y′ = y + 3
c. x′ = y
y′ = x
2. Answer the following questions about the transformation x′ = x + 2y, y′ = 3x – y.
a. Find the image of the point (4, 5).
b. Find a point (x, y) whose image is the point (5, 8). Hint: Set x′ = 5 and y′ = 8 to get
a system of linear equations that can be solved.
Name
June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 2
3. Apply each of the following transformations to the given “house” figure. (You can do this
by transforming points A through F, then connecting the dots.) Then, describe the
transformation using geometric words.
a. x′ = x – 7, y′ = y + 3
b. x′ = –x, y′ = y
Name
June 3/4, 2015
c. x′ = 2x, y′ = –y
d. x′ = –y, y′ = x
Math 4 notes and problems
Transformation rules and matrices page 3
Name
June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 4
Name
June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 5
Transformation matrices
A linear transformation is a transformation whose rule has the form
x′ = ax + by,
y′ = cx + dy.
Such a rule can always be rewritten as a matrix multiplication equation:
⎡a b ⎤ ⎡ x ⎤ ⎡ x ʹ′⎤
⎢ c d ⎥ ⋅ ⎢ y ⎥ = ⎢ y ʹ′⎥ .
⎣
⎦ ⎣ ⎦ ⎣ ⎦
The 2-by-2 matrix is called the transformation matrix for this transformation.
The useful thing about rewriting a transformation rule using a transformation matrix is that
image points can now be calculated using matrix multiplication.
Example
Consider the transformation from problem 2: x′ = x + 2y, y′ = 3x – y.
The rule can be written as
⎡1 2 ⎤ ⎡ x ⎤ ⎡ x ʹ′ ⎤
⎢3 − 1⎥ ⋅ ⎢ y ⎥ = ⎢ y ʹ′⎥ .
⎣
⎦ ⎣ ⎦ ⎣ ⎦
You can find the image of point (4, 5) by doing this matrix multiplication:
⎡1 2 ⎤ ⎡4⎤
⎢3 − 1⎥ ⋅ ⎢5⎥ =
⎣
⎦ ⎣ ⎦
It gets better: you can calculate the images of all the vertex points of the “house” figure
using one big matrix multiplication!
⎡1 2 ⎤ ⎡2 2 4 5 6 6⎤
⎢3 − 1⎥ ⋅ ⎢1 3 3 4 3 1⎥ =
⎣
⎦ ⎣
⎦
Name
June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 6
Problems
4. Here is a transformation rule written using a matrix:
⎡− 2 0⎤ ⎡ x ⎤ ⎡ x ʹ′ ⎤
⎢ 0 1⎥ ⋅ ⎢ y ⎥ = ⎢ y ʹ′⎥ .
⎣
⎦ ⎣ ⎦ ⎣ ⎦
a. Calculate the image of the point (5, 4).
b. Calculate the image of the point (a, b).
c. Calculate and draw the image of this figure. You can use your calculator to do the matrix
multiplication.
d. Describe the transformation using geometric words.
Name
June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 7
5. Here is a transformation rule written using a matrix:
⎡0 1⎤ ⎡ x ⎤ ⎡ xʹ′ ⎤
⎢1 0⎥ ⋅ ⎢ y ⎥ = ⎢ yʹ′⎥ .
⎣
⎦ ⎣ ⎦ ⎣ ⎦
a. Calculate the image of the point (5, 4).
b. Calculate the image of the point (a, b).
c. Calculate and draw the image of this figure. You can use your calculator to do the matrix
multiplication.
d. Describe the transformation using geometric words.
Name
June 3/4, 2015
Math 4 notes and problems
Transformation rules and matrices page 8
6. For each transformation, write a transformation rule as a pair of equations.
a. reflection across the y-axis
x′ =
y′ =
b. translation down by 5 and left by 1
x′ =
y′ =
c. stretching horizontally by a factor of 2 and shrinking vertically by a factor of
1
2
x′ =
y′ =
7. For each transformation, write a transformation rule as a matrix multiplication equation.
a. stretching by a factor of 3 both horizontally and vertically
b. reflecting across the x-axis and stretching vertically by a factor of 5
c. reflecting across the x-axis and stretching horizontally by a factor of 5
8. The goal of this problem is to figure out what transformation is represented by the matrix
⎡ 0 − 1⎤
⎢− 1 0 ⎥
⎣
⎦
a. On graph paper, draw a figure made up by you that is comparable to my “house.”
(Choose a figure that has about 6-to-8 vertex points.)
b. Apply the transformation to your figure. (You may use your calculator for the matrix
multiplication.) Draw the image figure.
c. Describe the transformation geometrically.