Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 23: Modeling Video Game Motion with Matrices
Student Outcomes

Students use matrix transformations to model circular motion.

Students use coordinate transformations to represent a combination of motions.
Lesson Notes
Students have recently learned how to represent rotations as matrix transformations. In this lesson, they apply that
knowledge to represent dynamic motion, as seen in video games. Students analyze circular motion that involves a time
𝜋
2
parameter such as 𝐺(𝑡) = (
𝜋
sin ( ∙ 𝑡)
2
cos ( ∙ 𝑡)
𝜋
2
). The second part of the lesson involves modeling a combination
𝜋
cos ( ∙ 𝑡)
2
− sin ( ∙ 𝑡)
of motions. For instance, students model motion along a circle followed by a translation or motion along a line followed
by a translation.
Classwork
The Opening Exercise allows students to practice matrix transformations and plot the results. This prepares students for
skills needed in this lesson. Work through this as a whole class, asking questions to assess student understanding.
Use this as a way to clear up misconceptions.
Opening Exercise (5 minutes)
Opening Exercise
𝝅
𝝅
𝟑
𝟑
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝒙
𝒙
𝟑
𝟑
Let 𝑹 (𝒚) = (
𝝅
𝝅 ) (𝒚).
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
a.
𝒙
𝒙
Describe the geometric effect of performing the transformation (𝒚) → 𝑹 (𝒚).
Applying 𝑹 rotates each point in the plane about the origin through
b.
𝝅
𝟑
radians in a counterclockwise direction.
𝟏
𝟏
Plot the point ( ), and then find 𝑹 ( ) and plot it.
𝟎
𝟎
𝝅
𝐜𝐨𝐬 ( )
𝟏
𝟑
𝑹( ) = (
𝝅 )
𝟎
𝐬𝐢𝐧 ( )
𝟑
Lesson 23:
Modeling Video Game Motion with Matrices
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
c.
𝟏
If we want to show that 𝑹 has been applied twice to (𝟏, 𝟎), we can write 𝑹𝟐 ( ), which represents
𝟎
𝟏
𝟏
𝟏
𝟏
𝑹 (𝑹 ( )). Find 𝑹𝟐 ( ) and plot it. Then, find 𝑹𝟑 ( ) = 𝑹 (𝑹 (𝑹 ( ))), and plot it.
𝟎
𝟎
𝟎
𝟎
𝟐𝝅
𝐜𝐨𝐬 ( )
𝐜𝐨𝐬(𝝅)
𝟏
𝟏
𝟑
𝑹𝟐 ( ) = (
)
); 𝑹𝟑 ( ) = (
𝟐𝝅
𝐬𝐢𝐧(𝝅)
𝟎
𝟎
𝐬𝐢𝐧 ( )
𝟑
d.
𝒙
𝒙
Describe the matrix transformation (𝒚) → 𝑹𝟐 (𝒚) using a single matrix.
𝒙
𝒙
𝝅
𝑹𝟐 (𝒚) is the transformation that rotates points through 𝟐 ∙ radians, so a formula for 𝑹𝟐 (𝒚) is
MP.7
𝟐𝝅
𝟐𝝅
𝟑
𝟑
𝟑
𝐜𝐨𝐬 ( ) − 𝐬𝐢𝐧 ( ) 𝒙
𝟑
𝟑
(
) (𝒚).
𝟐𝝅
𝟐𝝅
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
Discussion (3 minutes): Circular Motion over Time
Let 𝑅(𝑡) = (
cos⁡(𝑡)
sin⁡(𝑡)
−sin⁡(𝑡) 1
) ( ).
cos⁡(𝑡) 0
Suppose that 𝑡 is measured in degrees. Let’s place several input-output pairs for this function on a graph:


𝑅(30) = (0.87,0.50)
𝑅(90) = ⁡ (0,1)
Lesson 23:
𝑅(45) = (0.71,0.71)
𝑅(180) = ⁡ (−1,0)
𝑅(60) = (0.50,0.87)
𝑅(270) = ⁡ (0, −1)
Modeling Video Game Motion with Matrices
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𝑅(360) = ⁡ (1,0)
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS

What do you notice about the points on the graph?


The points appear to lie on a circle.
How could we be sure the points are actually on a circle?


If each point is the same distance from the origin, then the points form a circle.
Now check and see if this is, in fact, the case. Have different students find the distance to the origin from given
points.

Students check the distance from a given point to the origin and confirm using the distance formula.
For example, 𝑅(30) = (0.87, 0.50). Its distance from the origin is √(0.87 − 0)2 + (0.5 − 0)2 = 1.

What is the result of R(t)?
cos⁡(𝑡) −sin⁡(𝑡) 1
cos(𝑡)
 𝑅(𝑡) = (
)( ) = (
)
sin⁡(𝑡) cos⁡(𝑡) 0
sin(𝑡)

Does this result ensure points lie on a circle?
Yes, this would confirm that the points lie on the unit circle because the 𝑥-value⁡corresponds to cosine
of an angle 𝑡 and the 𝑦-value corresponds to sine of the same angle on the unit circle.

Exercise 1 (4 minutes)
This exercise provides students more practice with matrices representing rotations. This time, the angle is different in
each function, allowing them to compare the results. Give students time to work on the following problems
independently; then, call on students to share their responses with the class.
Exercises
1.
𝒕
𝒕
𝟐
𝟐
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝟏
𝐜𝐨𝐬⁡(𝟐𝒕) −𝐬𝐢𝐧(𝟐𝒕) 𝟏
𝟐
𝟐
Let 𝒇(𝒕) = (
) ( ), and let 𝒈(𝒕) = (
𝒕
𝒕 ) (𝟎).
𝐬𝐢𝐧(𝟐𝒕) 𝐜𝐨𝐬(𝟐𝒕) 𝟎
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
a.
Suppose 𝒇(𝒕) represents the position of a moving object that starts at (𝟏, 𝟎). How long does it take for this
object to return to its starting point? When the argument of the trigonometric function changes from 𝒕 to 𝟐𝒕,
what effect does this have?
The object will return to (𝟏, 𝟎) when 𝟐𝒕 = 𝟐𝝅. Thus, it will take 𝒕 = 𝝅 seconds for this to happen. Changing
the argument from 𝒕 to 𝟐𝒕 causes the object to move twice as fast.
b.
If the position is given instead by 𝒈(𝒕),⁡how long would it take the object to return to its starting point?
𝒕
When the argument of the trigonometric function changes from 𝒕 to , what effect does this have?
𝟐
The object will return to (𝟏, 𝟎) when
Changing the argument from 𝒕 to
Lesson 23:
𝒕
𝟐
𝒕
𝟐
= 𝟐𝝅. Thus, it will take 𝒕 = 𝟒𝝅 seconds for this to happen.
causes the object to move half as fast.
Modeling Video Game Motion with Matrices
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Example 1 (4 minutes)
cos⁡(𝑡) −sin⁡(𝑡) 3
Let 𝐹(𝑡) = (
) ( ).
sin⁡(𝑡) cos⁡(𝑡) 2

𝜋
2
This time, we’ll measure 𝑡 in radians. Find 𝐹 ( ), 𝐹(𝜋), 𝐹 (
𝜋
2
𝜋
2
𝜋
sin ( )
2
𝜋
3
0
2
𝜋 ) ( 2) = ( 1
cos ( )⁡
2
cos ( )
− sin ( )

𝐹 ( ) =⁡(

𝐹(𝜋) = ⁡ (

3𝜋
cos ( )
3𝜋
2
𝐹 ( ) =⁡(
3𝜋
2
sin ( )
2
cos(π)
sin(𝜋)
− sin(𝜋) 3
−1
)( ) = (
2
0
cos(𝜋)
3𝜋
), and 𝐹(2𝜋).
2
0−2
−1 3
−2
)( ) = (
)=( )
3+0
0
2
3
0
3
−3 + 0
−3
)( ) = (
)=( )
−1 2
0−2
−2
3𝜋
) 3
0
2
)( ) = (
3𝜋
2
−1
cos ( )⁡
2
− sin (
0+2
1 3
2
)( ) = (
)=( )
−3 + 0
0 2
−3
cos(2π) − sin(2𝜋) 3
3+0
1 0 3
3
𝐹(2𝜋) = ⁡ (
)( ) = (
)( ) = (
)=( )
0+2
2
0 1 2
2
sin(2𝜋) cos(2𝜋)
When we plot points, we see once again that they appear to lie on a circle. Make sure this is really true.
cos⁡(𝑡) −sin⁡(𝑡) 3
3 cos(𝑡) − 2 sin(𝑡)
 𝐹(𝑡) = (
)( ) = (
)
sin⁡(𝑡) cos⁡(𝑡) 2
3 sin(𝑡) + 2 cos(𝑡)


(3 cos(𝑡) − 2 sin(𝑡))2 + ( 3 sin(𝑡) + 2 cos(𝑡))2
9 cos 2(𝑡) − 12 cos(𝑡) sin(𝑡) + 4 sin2 (𝑡)⁡ + 9 sin2 (𝑡) + 12 cos(𝑡) sin(𝑡) + 4 cos 2(𝑡)
9(cos 2(𝑡) + sin2 (𝑡)) + 4(sin2 (𝑡) + cos 2 (𝑡))
9(1) + 4(1) = 9 + 4 = 13
Thus, each point is √13 units from the origin, which confirms that the outputs lie on a circle.
Lesson 23:
Modeling Video Game Motion with Matrices
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23
M1
PRECALCULUS AND ADVANCED TOPICS
Discussion (4 minutes): Rotations That Use a Time Parameter
𝜋
2
Let 𝐹(𝑡) = (
𝜋
sin ( ∙ 𝑡)
2
cos ( ∙ 𝑡)


Draw the path that 𝑃 = 𝐹(𝑡) traces out as 𝑡 varies within each of the following intervals:

0≤𝑡≤1

1≤𝑡≤2

2≤𝑡≤3

3≤𝑡≤4
Where will the object be located at 𝑡 = 0.5 second?


𝜋
1
2
) ( ).
𝜋
0
cos ( ∙ 𝑡)
2
− sin ( ∙ 𝑡)
𝜋
4
𝜋
4
𝑓(0.5) = (cos ( ) , sin ( )) ≈ (0.71,0.71)
How long will it take the object to reach (0.71, −0.71)?

These coordinates represent (cos (
location at 𝑡 =
7
= 3.5 seconds.
2
7𝜋
7𝜋
𝜋 7
𝜋 7
) , sin ( )), so (cos ( ∙ ) , sin ( ∙ )). The object reaches this
4
4
2 2
2 2
Exercises 2–3 (5 minutes)
Give students time to complete the following exercises; then, ask them to compare their responses with a partner.
Call on students to share their responses with the class.
Lesson 23:
Modeling Video Game Motion with Matrices
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
2.
𝝅
𝝅
𝟐
𝟐
𝐜𝐨𝐬 ( ∙ 𝒕) −𝐬𝐢𝐧 ( ∙ 𝒕) 𝟎
𝟐
𝟐
Let 𝑮(𝒕) = (
) ( ).
𝝅
𝝅
𝟏
𝐬𝐢𝐧 ( ∙ 𝒕) 𝐜𝐨𝐬 ( ∙ 𝒕)
a.
Draw the path that 𝑷 = 𝑮(𝒕) traces out as 𝒕 varies within the interval 𝟎 ≤ 𝒕 ≤ 𝟏.
b.
Where will the object be at 𝒕 = 𝟑 seconds?
𝑮(𝟑) = (𝟏, 𝟎)
c.
How long will it take the object to reach (𝟎, −𝟏)?
𝝅
𝟐
𝝅
𝟐
These coordinates represent (𝐜𝐨𝐬(𝝅) , 𝐬𝐢𝐧(𝝅)), so (𝐜𝐨𝐬 ( ∙ 𝟐) , 𝐬𝐢𝐧 ( ∙ 𝟐)); the object reaches this location
at 𝒕 = 𝟐 seconds. 𝑮(𝟐) = (𝟎, −𝟏), so it will take 𝟐 seconds to reach that location.
3.
𝝅
𝝅
𝟐
𝟐
𝐜𝐨𝐬 ( ∙ 𝒕) −𝐬𝐢𝐧 ( ∙ 𝒕) 𝟏
𝟐
𝟐
Let 𝑯(𝒕) = (
) ( ).
𝝅
𝝅
𝟒
𝐬𝐢𝐧 ( ∙ 𝒕) 𝐜𝐨𝐬 ( ∙ 𝒕)
a.
Draw the path that 𝑷 = 𝑯(𝒕) traces out as 𝒕 varies within the interval 𝟎 ≤ 𝒕 ≤ 𝟐.
b.
Where will the object be at 𝒕 = 𝟏 second?
𝑯(𝟏) = (−𝟒, 𝟏)
c.
How long will it take the object to return to its starting point?
𝑯(𝟒) = (𝟏, 𝟒), so it will take 𝟒 seconds to return to its starting point.
Lesson 23:
Modeling Video Game Motion with Matrices
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Example 2 (4 minutes)
𝜋
6
Let 𝑓(𝑡) = (
𝜋
sin ( ∙ 𝑡)
6
cos ( ∙ 𝑡)


𝜋
6
6
) ( ).
𝜋
2
cos ( ∙ 𝑡)
6
− sin ( ∙ 𝑡)
Draw the path that 𝑃 = 𝑓(𝑡) traces out as 𝑡 varies within each of the following intervals:

0≤𝑡≤1
1≤𝑡≤2
2≤𝑡≤3

3≤𝑡≤4
4≤𝑡≤5
5≤𝑡≤6
As an example, can you describe what happens to the object as 𝑡 varies within the interval 0 ≤ 𝑡 ≤ 1?

Since 𝑓(0) = (6,2), the object starts its trajectory there. When 𝑡 = 1, the object will have moved
through
𝜋
6
radians. So, in the time interval 0 ≤ 𝑡 ≤ 1, the object moves along a circular arc as shown
below.
Exercises 4–5 (3 minutes)
Give students time to complete the following exercises; then ask them to compare their responses with a partner.
Call on students to share their responses with the class, and use this as an opportunity to check for understanding.
4.
Suppose you want to write a program that takes the point (𝟑, 𝟓) and rotates it about the origin to the point
(−𝟑, −𝟓) over a 𝟏-second interval. Write a function 𝑷 = 𝒇(𝒕) that encodes this rotation.
𝐜𝐨𝐬⁡(𝝅 ∙ 𝒕) −𝐬𝐢𝐧⁡(𝝅 ∙ 𝒕) 𝟑
Let 𝒇(𝒕) = (
) ( ). We have 𝒇(𝟎) = (𝟑, 𝟓) and 𝒇(𝟏) = (−𝟑, −𝟓), as required.
𝐬𝐢𝐧⁡(𝝅 ∙ 𝒕) 𝐜𝐨𝐬⁡(𝝅 ∙ 𝒕) 𝟓
5.
If instead you wanted the rotation to take place over a 𝟏. 𝟓-second interval, how would your function change?
𝒕
𝒕
𝟏
𝟏
𝐜𝐨𝐬 (𝝅 ∙ ∙ 𝟓) −𝐬𝐢𝐧 (𝝅 ∙ ∙ 𝟓) 𝟑
𝟏
𝟏
Let 𝒇(𝒕) = (
) ( ).⁡⁡We have 𝒇(𝟎) = (𝟑, 𝟓) and 𝒇(𝟏. 𝟓) = (−𝟑, −𝟓), as required.
𝒕
𝒕
𝟓
𝐬𝐢𝐧 (𝝅 ∙ ∙ 𝟓) 𝐜𝐨𝐬 (𝝅 ∙ ∙ 𝟓)
Lesson 23:
Modeling Video Game Motion with Matrices
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Example 3 (4 minutes)
𝜋
2
Let’s analyze the transformation 𝑔(𝑡) = (
𝜋
3 sin ( ∙ 𝑡)
2
3 cos ( ∙ 𝑡)
and 𝑔(1).
𝜋
1
2
) ( ). In particular, we will compare 𝑔(0)
𝜋
2
3 cos ( ∙ 𝑡)
2
−3 sin ( ∙ 𝑡)

What is 𝑔(0)? What geometric effect does 𝑔(𝑡) have on (1, 2) initially?

What is 𝑔(1)? Describe what is going on.



We have 𝑔(3) = (6, −3), which represents a quarter turn of the point (−3, −6) about the origin in a
counterclockwise direction.
What is 𝑔(4)? Describe what is going on.


We have 𝑔(2) = (−3, −6), which represents a quarter turn of the point (−6, 3) about the origin in a
counterclockwise direction.
What is 𝑔(3)? Describe what is going on.


This transformation combines a quarter turn about the origin with a scaling by a factor of 3.
What is 𝑔(2)? Describe what is going on.


We have 𝑔(1) = (−6, 3), which represents a quarter turn of the point (3, 6) about the origin in a
counterclockwise direction.
Can you summarize the geometric effect of applying 𝑔(𝑡) to the point (1, 2) during the time interval
0 ≤ 𝑡 ≤ 1?


We have 𝑔(0) = (3, 6), which is a dilation of (1, 2) using scale factor 3.
We have 𝑔(1) = (3, 6), which represents a quarter turn of the point (6, −3) about the origin in a
counterclockwise direction.
Compare 𝑔(0) and 𝑔(4). Does this make sense?

𝑔(0) = 𝑔(4) This makes sense because 4 quarter turns would be a full rotation, so this would bring
you back to the starting point.
Lesson 23:
Modeling Video Game Motion with Matrices
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23
M1
PRECALCULUS AND ADVANCED TOPICS
Closing (4 minutes)

Write one to two sentences in your notebook describing what you learned in today’s lesson; then, share your
response with a partner.

We learned how to use matrices to describe rotations that happen over a specific time interval. We
also discussed how to model multiple transformations, such as a rotation followed by a translation.
Exit Ticket (5 minutes)
Lesson 23:
Modeling Video Game Motion with Matrices
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Name
Date
Lesson 23: Modeling Video Game Motion with Matrices
Exit Ticket
Write a function 𝑓(𝑡) that incorporates the following actions. Make a drawing of the path the point follows during the
time interval 0 ≤ 𝑡 ≤ 3.
a.
During the time interval 0 ≤ 𝑡 ≤ 1, move the point (8, 6) through
𝜋
4
radians about the origin in a
counterclockwise direction.
b.
During the time interval 1 < 𝑡 ≤ 3, move the image along a straight line to (6, −8).
Lesson 23:
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
318
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Exit Ticket Sample Solutions
Write a function 𝒇(𝒕) that incorporates the following actions. Make a drawing of the path the point follows during the
time interval 𝟎 ≤ 𝒕 ≤ 𝟑.
a.
During the time interval 𝟎 ≤ 𝒕 ≤ 𝟏, move the point (𝟖, 𝟔) through
𝝅
𝟒
radians about the origin in a
counterclockwise direction.
⁡𝝅𝒕⁡
⁡𝝅𝒕⁡
𝐜𝐨𝐬 (
)⁡ −𝐬𝐢𝐧 (
)
𝟖
⁡𝟒⁡
⁡𝟒⁡
𝒇(𝒕) = (
⁡𝝅𝒕⁡
⁡𝝅𝒕⁡ ) (𝟔) ,⁡⁡⁡⁡⁡⁡𝟎 ≤ 𝒕 ≤ 𝟏
𝐬𝐢𝐧 (
) 𝐜𝐨𝐬 (
)
⁡𝟒⁡
⁡𝟒⁡
𝐜𝐨𝐬(𝟎)⁡ −𝐬𝐢𝐧(𝟎) 𝟖
𝟏⁡ 𝟎 𝟖
𝟖
𝒇(𝟎) = (
)( ) = (
)( ) = ( )
𝐬𝐢𝐧(𝟎) 𝐜𝐨𝐬(𝟎) 𝟔
𝟎 𝟏 𝟔
𝟔
⁡𝝅⁡
⁡𝝅⁡
𝐜𝐨𝐬 ( )⁡ −𝐬𝐢𝐧 ( )
𝟖
⁡𝟒⁡
⁡𝟒⁡
𝒇(𝟏) = (
⁡𝝅⁡
⁡𝝅⁡ ) (𝟔) =
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
⁡𝟒⁡
⁡𝟒⁡
b.
√𝟐 −√𝟐
⁡
𝟖
𝟏. 𝟒𝟏
𝟐
𝟐
( ) = ( √𝟐 ) ⁡ ≈ (
)⁡
𝟔
𝟗. 𝟗𝟎
𝟕√𝟐
√𝟐 √𝟐
𝟐 )
( 𝟐
During the time interval 𝟏 < 𝒕 ≤ 𝟑, move the image along a straight line to (𝟔, −𝟖).
𝟔
The image is ( √𝟐 ) → ( ) in 𝟐 seconds from 𝟏 < 𝒕 ≤ 𝟑.
−𝟖
𝟕√𝟐
√𝟐 − 𝒌𝒕 = 𝟔
⁡√𝟕𝟐 − 𝒎𝒕 = −𝟖
√𝟐 − 𝟐𝒌 = 𝟔
𝟕√𝟐 − 𝟐𝒎 = −𝟖
𝒌=
𝒉(𝒕) =
√𝟐 − 𝟔⁡
𝟐
𝒎=
𝟕√𝟐 + 𝟖⁡
𝟐
(√𝟐 − 𝟔)𝒕⁡
𝟑
(𝟕√𝟐 + 𝟖)𝒕⁡
𝟕√𝟐 −
𝟑
(
)
√𝟐 −
Problem Set Sample Solutions
1.
⁡𝝅⁡
⁡𝝅⁡
𝟒
𝟒
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝒙
𝒙
𝟒
𝟒
Let 𝑹 (𝒚) = (
) (𝒚). Find the following.
⁡𝝅⁡
⁡𝝅⁡
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
a.
𝑹𝟐 (√𝟐)
√𝟐
𝝅
𝝅
√𝟐
√𝟐
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
−
√𝟐
√𝟐
√𝟐
𝟒
𝟒
𝟐
𝟐 (√𝟐)
𝑹 ( ) = 𝑹 (𝑹 ( )) = 𝑹 (
𝝅
𝝅 )( ) = 𝑹
√𝟐 √𝟐
√𝟐
√𝟐
√𝟐
√𝟐
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟒
𝟒
𝟐 )
(
)
(( 𝟐
)
𝟐
𝝅
𝝅
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
𝟎
𝟎
𝟒
𝟒
= 𝑹( ) = (
𝝅
𝝅 ) (𝟐) =
𝟐
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟒
𝟒
Lesson 23:
√𝟐
√𝟐
−
𝟐
𝟐 (𝟎) = (−√𝟐)
𝟐
√𝟐 √𝟐
√𝟐
𝟐 )
(𝟐
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
319
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
b.
How many transformations do you need to take so that the image returns to where it started?
It rotates by
𝝅
𝟒
radians for each transformation; therefore, it takes 𝟖 times to get to 𝟐𝝅, which is where it
started.
c.
𝒙
𝒙
𝒙
Describe the matrix transformation (𝒚) → 𝑹𝟐 (𝒚)⁡⁡and 𝑹𝒏 (𝒚) using a single matrix.
𝒙
𝒙
𝝅
𝑹𝟐 = (𝒚) is the transformation that rotates the point through 𝟐 × radian, so a formula for⁡𝑹𝟐 (𝒚) is
𝟒
𝟐𝝅
𝟐𝝅
𝝅
𝛑
𝝅
𝝅
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝒙
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝒙
𝒙 ∙ 𝐜𝐨𝐬 ( ) − 𝒚 ∙ 𝐬𝐢𝐧 ( )
𝟒
𝟒
𝟐
𝟐
𝟐
𝟐
(
) (𝒚) = (
𝝅
𝝅 ) (𝒚) = (
𝝅
𝝅 ).
𝟐𝝅
𝟐𝝅
𝐬𝐢𝐧
(
)
𝐜𝐨𝐬
(
)
𝒙
∙
𝐬𝐢𝐧
(
)
+
𝒚
∙
𝐜𝐨𝐬
(
)
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟐
𝟐
𝟐
𝟐
𝟒
𝟒
𝒏𝝅
𝒏𝝅
𝒙 ∙ 𝐜𝐨𝐬 ( ) − 𝒚 ∙ 𝐬𝐢𝐧 ( )
𝒙
𝟒
𝟒
⁡𝑹 = (𝒚) ⁡is ⁡ (
𝒏𝝅
𝒏𝝅 ) .
𝒙 ∙ 𝐬𝐢𝐧 ( ) + 𝒚 ∙ 𝐜𝐨𝐬 ( )
𝟒
𝟒
𝒏
2.
𝐜𝐨𝐬(𝒕) −𝐬𝐢𝐧(𝒕) 𝟏
𝟏
For 𝒇(𝒕) = (
) ( ), it takes 𝟐𝝅 to transform the object at ( ) back to where it starts. How long
𝐬𝐢𝐧(𝒕) 𝐜𝐨𝐬(𝒕) 𝟏
𝟏
does it take the following functions to return to their starting point?
a.
𝐜𝐨𝐬(𝟑𝒕) −𝐬𝐢𝐧(𝟑𝒕) 𝟏
𝒇(𝒕) = (
)( )
𝐬𝐢𝐧(𝟑𝒕) 𝐜𝐨𝐬(𝟑𝒕) 𝟏
𝟑𝒕 = 𝟐𝝅,
𝒕=
𝟐𝝅
𝟑
⁡𝒕⁡
b.
⁡𝒕⁡
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝟏
𝟑
𝟑
𝒇(𝒕) = (
) ( )⁡
⁡𝒕⁡
⁡𝒕⁡
𝟏
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟑
𝒕
= 𝟐𝝅,
𝟑
𝟑
𝒕 = 𝟔𝝅
⁡𝟐𝒕⁡
c.
⁡𝟐𝒕⁡
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝟏
𝟓
𝟓
𝒇(𝒕) = (
)( )
⁡𝟐𝒕⁡
⁡𝟐𝒕⁡
𝟏
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟓
𝟐𝒕
= 𝟐𝝅,
𝟓
Lesson 23:
𝟓
𝒕 = 𝟓𝝅
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
320
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
3.
𝐜𝐨𝐬(𝒕) −𝐬𝐢𝐧(𝒕) 𝟐
Let 𝑭(𝒕) = (
) ( ), where 𝒕⁡is measured in radians. Find the following:
𝐬𝐢𝐧(𝒕) 𝐜𝐨𝐬(𝒕) 𝟏
a.
𝑭(
⁡𝟑𝝅⁡
⁡𝟕𝝅⁡
),𝑭(
), and the radius of the path
𝟐
𝟔
𝟑𝝅
𝟑𝝅
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
𝟑𝝅
𝟐
𝟐 ) (𝟐) = ( 𝟎 𝟏) (𝟐) = ( 𝟏 )
𝑭( ) = (
𝟑𝝅
𝟑𝝅
−𝟏 𝟎 𝟏
𝟏
−𝟐
𝟐
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟐
𝟐
𝟕𝝅
𝟕𝝅
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
𝟕𝝅
𝟔
𝟔 ) (𝟐) =
𝑭( ) = (
𝟕𝝅
𝟕𝝅
𝟏
𝟔
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟔
𝟔
𝟏
𝟏
√𝟑
−√𝟑 +
𝟐
𝟐
𝟐
𝟐
( )=
√𝟑
𝟏
√𝟑 𝟏
−𝟏 −
−
−
(
𝟐)
𝟐)
( 𝟐
−
The path of the point from 𝟎 ≤ 𝒕 ≤ 𝟐𝝅 is a circle with a center at (𝟎, 𝟎).
Thus, the radius = √𝒙𝟐 + 𝒚𝟐 = √(𝟏)𝟐 + (𝟐)𝟐 = √𝟓⁡ or
𝟏
𝟐
𝟐
the radius = √𝒙𝟐 + 𝒚𝟐 = √(−√𝟑 + ) + (−𝟏 −
b.
√𝟑
𝟐
𝟐
) = √𝟓.
𝐜𝐨𝐬(𝒕) −𝐬𝐢𝐧(𝒕) 𝒙
Show that the radius is always √𝒙𝟐 + 𝒚𝟐 for the path of this transformation (𝒕) = (
) ( ).
𝐬𝐢𝐧(𝒕) 𝐜𝐨𝐬(𝒕) 𝒚
𝐜𝐨𝐬(𝒕) −𝐬𝐢𝐧(𝒕) 𝒙
𝒙 𝐜𝐨𝐬(𝒕) − 𝒚 𝐬𝐢𝐧(𝒕)
𝑭(𝒕) = (
)( ) = (
)
𝐬𝐢𝐧(𝒕) 𝐜𝐨𝐬(𝒕) 𝒚
𝒙 𝐬𝐢𝐧(𝒕) + 𝒚 𝐜𝐨𝐬(𝒕)
𝟐
⁡The radius = √(𝒙 𝐜𝐨𝐬(𝒕) − 𝒚 𝐬𝐢𝐧(𝒕))𝟐 + (𝒙 𝐬𝐢𝐧(𝒕) + 𝒚 𝐜𝐨𝐬(𝒕))
= √𝒙𝟐 𝐜𝐨𝐬 𝟐 (𝒕) − 𝟐𝒙𝒚 𝐜𝐨𝐬(𝒕)𝐬𝐢𝐧(𝒕) + 𝒚𝟐 𝐬𝐢𝐧𝟐 (𝒕) + 𝒙𝟐 𝐬𝐢𝐧𝟐 (𝒕) + 𝟐𝒙𝒚 𝐬𝐢𝐧(𝒕)𝐜𝐨𝐬(𝒕) + 𝒚𝟐 𝐜𝐨𝐬 𝟐 (𝒕)⁡
= √𝒙𝟐 𝐜𝐨𝐬 𝟐 (𝒕) + 𝒙𝟐 𝐬𝐢𝐧𝟐 (𝒕) + 𝒚𝟐 𝐬𝐢𝐧𝟐 (𝒕) + 𝒚𝟐 𝐜𝐨𝐬𝟐 (𝒕)
= √𝒙𝟐 (𝐜𝐨𝐬 𝟐 (𝒕) + 𝐬𝐢𝐧𝟐 (𝒕)) + 𝒚𝟐 (𝐜𝐨𝐬𝟐 (𝒕) + 𝐬𝐢𝐧𝟐 (𝒕))⁡
= √𝒙𝟐 + 𝒚𝟐 .
⁡𝝅𝒕⁡
4.
⁡𝝅𝒕⁡
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( ) 𝟒
𝟐
𝟐
Let 𝑭(𝒕) = (
) ( ), where 𝒕⁡is a real number.
⁡𝝅𝒕⁡
⁡𝝅𝒕⁡
𝟒
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟐
a.
𝟐
Draw the path that 𝑷 = 𝑭(𝒕) traces out as 𝒕 varies within each of the following intervals:
i.
𝟎 ≤ 𝒕 ≤ 𝟏⁡⁡
ii.
𝟏 ≤ 𝒕 ≤ 𝟐⁡⁡
iii.
𝟐 ≤ 𝒕 ≤ 𝟑⁡⁡
iv.
𝟑 ≤ 𝒕 ≤ 𝟒⁡⁡
Lesson 23:
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
321
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
b.
Where will the object be located at 𝒕 = 𝟐. 𝟓 seconds?
𝟓𝝅
𝟓𝝅
−√𝟐 √𝟐
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
𝟎
𝟒
𝟒
𝟒
𝟒
𝟐
𝟐
𝑭(𝟐. 𝟓) = (
)( ) =
( )=(
)
𝟓𝝅
𝟓𝝅
𝟒
−𝟒√𝟐
−√𝟐 −√𝟐 𝟒
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟒
𝟒
𝟐 )
( 𝟐
c.
−𝟖√𝟔
How long does it take the object to reach (
)?
𝟖√𝟐
𝝅𝒕
⁡𝟖√𝟐⁡
𝝅 𝝅𝒕
𝝅
𝟏
−𝟖√𝟔⁡
The point (
) is in Quadrant II; the reference angle is
= 𝐚𝐫𝐜𝐭𝐚𝐧 (
) = 𝟔, 𝟐 = 𝟔,𝒕 = 𝟑
𝟐
𝟖√𝟔
𝟖√𝟐⁡
second.
It takes 𝟏. 𝟓 seconds to rotate the point to 𝝅; therefore, 𝟏. 𝟓 −
⁡𝝅𝒕⁡
5.
⁡𝟏⁡ ⁡𝟕⁡
= second.
𝟑
𝟔
⁡𝝅𝒕⁡
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
−𝟏
𝟑
𝟑
Let 𝑭(𝒕) = (
).
)(
⁡𝝅𝒕⁡
⁡𝝅𝒕⁡
−√𝟑
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟑
𝟑
a.
Draw the path that 𝑷 = 𝑭(𝒕) traces out as 𝒕 varies within the interval 𝟎 ≤ 𝒕 ≤ 𝟏.
b.
How long does it take the object to reach (√𝟑, 𝟎)?
The point (√𝟑, 𝟎) lies on the 𝒙-axis. Therefore, 𝒕 = 𝟐 seconds to rotate to the point (−𝟏, −√𝟑).
c.
How long does it take the object to return to its starting point?⁡
It takes 𝟔⁡seconds.
6.
Find the function that will rotate the point (𝟒, 𝟐) about the origin to the point (−𝟒, −⁡𝟐) over the following time
intervals.
a.
Over a 𝟏-second interval
𝐜𝐨𝐬(𝝅𝒕) −𝐬𝐢𝐧(𝝅𝒕) 𝟒
𝒇(𝒕) = (
)( )
𝐬𝐢𝐧(𝝅𝒕) 𝐜𝐨𝐬(𝝅𝒕) 𝟐
b.
Over a 𝟐-second interval
𝝅𝒕
𝝅𝒕
𝐜𝐨𝐬 ( ) −𝐬𝐢𝐧 ( )
𝟒
𝟐
𝟐
𝒇(𝒕) = (
𝝅𝒕
𝝅𝒕 ) (𝟐)
𝐬𝐢𝐧 ( ) 𝐜𝐨𝐬 ( )
𝟐
𝟐
Lesson 23:
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
322
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
c.
𝟏
Over a -second interval
𝟑
𝐜𝐨𝐬(𝟑𝝅𝒕) −𝐬𝐢𝐧(𝟑𝝅𝒕) 𝟒
𝒇(𝒕) = (
)( )
𝐬𝐢𝐧(𝟑𝝅𝒕) 𝐜𝐨𝐬(𝟑𝝅𝒕) 𝟐
d.
𝟒
How about rotating it back to where it starts over a -second interval?
𝟓
𝟓𝝅𝒕
𝟓𝝅𝒕
𝐜𝐨𝐬 (
) −𝐬𝐢𝐧 (
)
𝟐
𝟐 ) (𝟒)
𝒇(𝒕) = (
𝟓𝝅𝒕
𝟓𝝅𝒕
𝟐
𝐬𝐢𝐧 (
) 𝐜𝐨𝐬 (
)
𝟐
𝟐
7.
Summarize the geometric effect of the following function at the given point and the time interval.
⁡𝝅𝒕⁡
a.
⁡𝝅𝒕⁡
𝟓 𝐜𝐨𝐬 ( ) −𝟓 𝐬𝐢𝐧 ( ) 𝟒
𝟒
𝟒
𝑭(𝒕) = (
)( ),𝟎 ≤ 𝒕 ≤ 𝟏
⁡𝝅𝒕⁡
⁡𝝅𝒕⁡
𝟑
𝟓 𝐬𝐢𝐧 ( ) 𝟓 𝐜𝐨𝐬 ( )
𝟒
𝟒
𝟒
𝟐𝟎
At 𝒕 = 𝟎, the point ( ) is dilated by a factor of 𝟓 to ( ).
𝟑
𝟏𝟓
𝝅
𝟐𝟎
At 𝒕 = 𝟏, the image ( ) then is rotated by radians counterclockwise about the origin.
𝟏𝟓
𝟒
b.
𝟏
⁡𝝅𝒕⁡
𝐜𝐨𝐬 ( )
𝟐
𝟔
𝑭(𝒕) = (
𝟏
⁡𝝅𝒕⁡
𝐬𝐢𝐧 ( )
𝟐
𝟔
𝟏
⁡𝝅𝒕⁡
) 𝟔
𝟐
𝟔
)( ),𝟎 ≤ 𝒕 ≤ 𝟏
𝟏
⁡𝝅𝒕⁡
𝟐
𝐜𝐨𝐬 ( )
𝟐
𝟔
− 𝐬𝐢𝐧 (
𝟏
𝟔
𝟑
At 𝒕 = 𝟎, the point ( ) is dilated by a factor of to ( ).
𝟐
𝟏
𝟐
𝝅
𝟑
At 𝒕 = 𝟏, the image ( ) then is rotated by radians counterclockwise about the origin.
𝟏
𝟔
8.
In programming a computer video game, Grace coded the changing location of a rocket as follows:
𝒙
At the time 𝒕 second between 𝒕 = 𝟎 seconds and 𝒕 = 𝟒 seconds, the location (𝒚) of the rocket is given by
𝝅
𝝅
𝐜𝐨𝐬 ( 𝒕)⁡ −𝐬𝐢𝐧 ( 𝒕)
√𝟐
𝟒
𝟒
(
𝝅
𝝅 )( ).
√𝟐
𝐬𝐢𝐧 ( 𝒕) 𝐜𝐨𝐬 ( 𝒕)
𝟒
𝟒
At a time of 𝒕 seconds between 𝒕 = 𝟒 and 𝒕 = 𝟖 seconds, the location of the rocket is given by
√𝟐
⁡(𝒕 − 𝟒)
𝟐
.
√𝟐
−√𝟐 +
⁡(𝒕 − 𝟒)
𝟐
(
)
−√𝟐 +
a.
What is the location of the rocket at time 𝒕 = 𝟎? What is its location at time 𝒕 = 𝟖?
𝐜𝐨𝐬(𝟎) −𝐬𝐢𝐧(𝟎)⁡ √𝟐
𝟏 𝟎 √𝟐
At 𝒕 = 𝟎, (
)( ) = (
) ( ) = (√𝟐).
𝐬𝐢𝐧(𝟎) 𝐜𝐨𝐬(𝟎)
𝟎 𝟏 √𝟐
√𝟐
√𝟐
√𝟐
−√𝟐 + (𝟖 − 𝟒)
−√𝟐 + 𝟐√𝟐
𝟐
At 𝒕 = 𝟒, (
)=(
) = (√𝟐).
√𝟐
−√𝟐
+
𝟐√𝟐
√𝟐
(𝟖
−√𝟐 +
− 𝟒)
𝟐
Lesson 23:
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
b.
Mason is worried that Grace may have made a mistake and the location of the rocket is unclear at time
𝒕 = 𝟒⁡ seconds. Explain why there is no inconsistency in the location of the rocket at this time.
𝐜𝐨𝐬(𝝅) −𝐬𝐢𝐧(𝝅)⁡ 𝟏
−𝟏 𝟎
−√𝟐
At 𝒕 = 𝟒, (
)( ) = (
) (√𝟐) = (
).
𝐬𝐢𝐧(𝝅) 𝐜𝐨𝐬(𝝅)
𝟎 −𝟏 √𝟐
𝟏
−√𝟐
√𝟐
−√𝟐 + (𝟒 − 𝟒)
−√𝟐
𝟐
At 𝒕 = 𝟒, (
)=(
).
√𝟐
−√𝟐
(𝟒
−√𝟐 +
− 𝟒)
𝟐
These are consistent.
c.
What is the area of the region enclosed by the path of the rocket from time 𝒕 = 𝟎 to 𝒕 = 𝟖?
The path traversed is a semicircle with a radius of 𝟐; the area enclosed is 𝑨 =
Lesson 23:
Modeling Video Game Motion with Matrices
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
𝝅𝒓𝟐 𝟒𝝅
=
= 𝟐𝝅 square units.
𝟐
𝟐
324
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