Download Secondary II H - Wasatch County School District

Sort and Classify
1
2
3
4
5
6
Write down as many ideas as you recall about objects that are congruent.
Include thoughts on how you might prove that two objects are congruent.
Write down as many ideas as you recall about objects that are similar.
Include thoughts on how you might prove that two objects are similar.
Use a compass to construct a segment congruent to the original. Use a compass to create a segment
similar to the original.
Write a complete sentence to describe how the compass can be used to accomplish these tasks.
7
Practice Set
1. Describe how your class sorted the shapes from the previous pages.
2. Use complete sentences to define congruence.
3. Write down the ideas about similar objects that surfaced in class discussion today.
4. Use complete sentences to describe how a compass can be used to create two congruent
segments.
Review
5.Use properties of equality to solve for x in each equation.
x
a. 5x 17  21
b.  8  3
6
c. 4x 12  9 x 11
6. Explain how someone can tell the difference between linear and exponential functions. If you
can’t remember, look it up and write down your notes.
7. To what mathematics can you connect today’s task?
Today’s task reminded me of … (topic and explanation).
8
Double the Logo
Your assignment is to double the logo drawn below.
Instructions:
Take two rubber bands of equal length and tie them together so that there are equally-sized bands on
either side of the knot. Pick and mark an anchor point somewhere on the paper. Pin the end of
rubber band to the anchor point with your finger. On the opposite side of the other band, place a
pen. Trace a new object while keeping the knot consistently on top of the figure you are trying to
enlarge.
Your task is to:
1. Follow directions as accurately as you can, but don’t worry if your pictures are wiggly.
2. Explain what in this procedure caused the shape to be twice the size of the original one. What
happens if you choose a different anchor point?
9
3. How can you perform the same process if instead of rubber bands you used a ruler? Try it out.
4. List as many different things you notice that
(a) stayed the same
(b) changed in this process.
10
Additional Space for Double the Logo
Questions and Notes
11
Before and After
Before: Use wikisticks, a ruler, and protractor to measure segments, arcs, and angles of circle O.
Notation:
Record all measurements in the table on the next page.
Create a point of dilation on the paper.
After: Use rubber bands and/or a ruler to create dilated images with the following scale factors:
2, 5/2, and 4. Record all measurements in the table.
12
AB
OE
ED
ACB
Perimeter
nD
EOB
Calculate
Perimeter
AB
Scale Factor
2
5/2
4
List as many different things you notice that
(a) stayed the same
(b) changed in this process.
13
Practice Set
1. Explain the class definition of dilation.
2. Explain what happens to a segment upon dilation.
3. What is the circumference of a circle?
4. Define the number pi. Do not list digits.
Review
5. Determine whether each set of data is linear, exponential, or neither. Explain how you know.
n
0
1
2
3
f(n)
5
10
20
40
n
0
1
2
3
f(n)
1
4
9
16
6. Use properties of equality to solve for x.
9 x  18
9x  18  27w
n
0
1
2
3
f(n)
21
19
17
15
tx  18  27w
7. To what mathematics can you connect today’s task?
Today’s task reminded me of … (topic and explanation).
14
Find the center
1. Rectangle A′B′C′D′ is the image of rectangle ABCD under a certain dilation. Find the
center of dilation P and the scale factor. Explain how you found it and why your method works.
2. The small rectangle A′B′C′D′ is the image of rectangle ABCD under a certain dilation.
Find the center of dilation P. What can you say about the scale factor?
15
3. Triangle A′B′C′ has the same shape as triangle ABC, but can't possibly be the image of
triangle ABC under any dilation. Explain why not.
4. Determine a sequence of transformations that will take the triangle ABC to triangle A′B′C′.
16
Notes for Finding the Center
17
As a group, answer the following three questions:
What is a scale factor?
How does the scale factor 2/5 compare with the scale factor 5/2?
How does the scale factor 2/5 compare with the scale factor 4/10?
What is a Rational Number?
What is not a Rational Number?
Notes on natural numbers, whole numbers, integers, real numbers.
18
Explain how to simplify each of the following operations on rational numbers.
Properties of Equality do not apply here.
Is the set of rational numbers closed over addition? Subtraction?
Is the set of rational numbers closed over multiplication? Division?
19
Practice Set
Write two sentences to compare your previous ideas of similarity with the ideas that we surfaced
today. How do these ideas compare with the class definition of dilation?
Draw a triangle A ' R ' T ' under the dilation with the given center and the scale factor:
a. Center O, scale factor ½
b. Center T, scale factor ½
c. Center O, scale factor 2
Use the figures below to answer these questions:
a. What is the scale factor that can be applied to figure B to get figure A?
b. What is the scale factor that can be applied to figure A to get figure B?
20
21
22
Dilations on a Coordinate System
A pool design originally is given as a quadrilateral with vertices A(1,1), B(3,5), C(6,8), D(0,4), The
owner wants it 3 times larger. Create the diagram of the original design and the dilated design.
Use coordinates to determine measurements and verify the scale.
Notes
23
Create a design with at least five key points. List the coordinates of these points.
Draw and describe the image after each operation as it compares with the original figure.
1. f ( x, y )  ( x  7, y  9)
2. f ( x, y )  ( y, x)
3. f ( x, y )  ( y,  x)
4. f ( x, y )  (4 x, 4 y )
5. f ( x, y )  ( x,3 y )
24
Notes:
Which of the previous transformations were congruent transformations?
Which of the previous transformations were dilation transformations?
Which of the previous transformations were neither congruent nor dilation transformations?
Write two sentences to describe how you will know what type of transformation is in use?
What is similarity?
25
From Here to There.
List the coordinates of points A, B, C, D, and E.
Cut out the shape below the grid and paste anywhere on the coordinate system.
List the coordinates of points A’, B’, C’, D’, and E’.
Trade with another student. Try to determine a sequence of transformations that map ABCDE to its
image. Use proper notation.
B
4
E
C
2
A
D
-10
-5
5
10
15
-2
-4
-6
Cut me out:
26
Given a triangle ABC with vertices A(-1,2), B(0,3), and C(5,-2), create a matrix of coordinates for
xB xC 
x
transformation. Use the format  A
 and call this matrix A.
y
y
y
B
C
 A
Perform each of the following operations. Use complete sentences to describe the transformation
caused by the operation.
1 0 
1. 
* A
0 1
3 0
2. 
* A
0 3
5 5 5
3. 
 A
 3 3 3
4. Create a matrix operation(s) that would reflect your triangle about the x-axis and dilate from the
origin by a scale factor of 7.
27
Practice Set
Name ________________________
Given and the scale factor, k, determine the coordinates of the dilated point,
the center of dilation is the origin.
. You may assume
1.
2.
3.
Given and the dilated point , find the scale factor. (What did you multiply the coordinates of
A by to get A’?). You may assume the center of dilation is the origin.
4.
5.
6.
For the given shapes, draw the dilation, given the scale factor and center.
7.
, center is
8.
, center is
9.
, center is
http://middlemathccss.wordpress.com
28
10. Explain the effects of the matrix operation on a matrix A of coordinates describing a polygon.
 0 1
 1 0  * A


11. Perform the following operations.
3 2

7 5
3 2

7 5
3 2

7 5
3 2

7 5
12. Create and label a point of dilation, D. Dilate the image with a scale factor of 1.5.
29
30
As a group, recall…
What are rational numbers?
What are not rational numbers?
Simplify each of the following and determine whether the number is rational or irrational.
8
81
18
1
Are there other sources/examples of irrational numbers besides the any found above?
Extra practice
31
32
Notes from Going Round in Circles
33
Practice Set
Write an equation of a circle for each center and radius or diameter measure given.
1. (0, 0) r = 2
2. (1, 1) r = 5
3. (-3, 2) d = 4
4. (-6, -1), d = 10
Find the coordinates of the center and the measure of the radius for each circle whose equation is
given.
5. ( x  5)2  ( y  2)2  49
6. ( x  3)2  ( y  7)2  100
7. x 2  ( y  4)2  7
Simplify each expression.
2
9. 7 
5
8. ( x  1)2  ( y  1)2  10
10.
4 1

7 9
11.
4 1

7 9
12. Choose a center of dilation and dilate the given figure using a scale factor of 1/3.
34
13. Label each of the following as linear, exponential, or neither. Explain how you know.
a. f (n)  14  3n
b. f (n)  90n2  1
c. f (n)  16  3n
d. f (n)  11  .4(n  1)
2
e. f (n)  1  ( ) n
3
f. 4 x  8 y  17
14. Determine the median and the IQR for the following data set:
17, 15, 14, 15, 18, 11, 12
15. Determine the y-intercept for the following functions.
a. y  15  3x
b. y  16n  3
c. y  14  2 x
35
36
37
Cut
38
39
Practice Set
Reduce each radical. Do write a decimal approximation. State whether each number is rational or
irrational.
a. 50
b. 88
c. 484
40
14. Use properties of equality to isolate the variable x.
x
 15
ax  b  c
t
x 8
h
w
15.
41
Cross Sections
Use clay to create several of each of the following shapes: cylinder, sphere, and cone. Use
the dental floss to cut a planar cross-section of each figure. For each slice, describe in words how
you cut the figure (be specific!) and sketch the planar cross-section. (The sketch should be only in
two dimensions.)
*Try several different angles to slice each shape.
Cylinders
Spheres
Cones
42
History and Information: Conic Sections
43
Rabbit Farm Task
Mary and Sam are going to start raising bunny rabbits to sell at Easter. They bought 24 feet of
fence to create a rectangular enclosure.
A. Draw the different enclosures Sam and Mary could make. Make sure to label the
dimensions of each rectangle.
B. Create multiple models to represent and organize the information from your rectangles.
C. Sam and Mary want to sell a lot of rabbits at Easter. Which one of the enclosures would
allow them to raise the most rabbits? Explain.
44
Planting a Garden Part I
1. John has an existing square garden in his backyard. He wants to increase the width by 4 feet and
the length by 6 feet.
A. What is the increase to the area of John’s garden?
B. Write a rule to describe the area of John’s garden.
x
x
2. Bob has a garden with a fence around it. He wants to maximize the garden’s area to get the most
out of the garden. He can’t afford to buy more fence.
A. What can Bob change to maximize the garden area?
B. Show and justify how your change created the largest possible garden.
x+7
x+3
45
Planting a Garden Part II
Describe the similarities and differences between Bob’s original garden and his new garden.
We developed 3 rules to describe Bob’s garden:
(x + 5 )2
(x + 5 ) (x + 5 )
x2 + 10x + 25
A. Show how these rules are the same and describe their differences.
B. Relate the similarities and differences of the rules back to the area model of Bob’s
garden.
46
Practice – More Gardens
1. James has a garden, shown below. What is the area of each part of the garden (ie: corn,
peas,…)?
x
3
x
Potatoes
Corn
4
Flowers
Peas
2. Susan knows the area of three sections of her garden, what is the area of the missing piece?
What are the dimensions of the entire garden? Area is measured in square feet.
Pumpkins
Squash
x2
6x
Green Beans
Onions
36
3. Write an expression that represents James’ (#1) and Susan’s (#2) gardens in more than one
form.
47
4. If the following rule x 2  8 x  15 is used to describe a garden, draw a model of the garden.
Make sure to label the dimensions and areas of each piece. Write the expression for this
garden in one other form.
For each,
Rewrite the expression in different forms.
( x  7)2
( x  14) 2
( x  4)2
3
( x  )2
2
In each circle equation, use multiplication to write the expressions in different forms.
( x  1)2  ( y  4)2  9
( x  8)2  ( y  3)2  25
( x  13)2  ( y  7)2  121
( x  4)2  ( y  6)2  8
48
How would we go from general conic equation to standard form?
Ex.
Add what you need to create a perfect square trinomial to the following expressions.
Then write in a different form.
x2 + 24x +
x2 +
x2 -30x +
x2 – 40x +
x2 
2
x
5
Show x 2  14 x  49 as the area of a square.
+ 49
1
x2  x 
3
Show x 2  10 x  25 as the area of a square.
49
Build perfect squares and use properties of equality to transform the general conic equation of a
circle into standard form. Then graph the circle.
1. x 2  8 x  y 2  2 y  64
2. x 2  y 2  14 x  12 y  4  0
3. x 2  2 x  y 2  55  10 y
50
Practice Set
1. Show x 2  24 x  144 as the area of a square.
2. Show x 2  30 x  225 as the area of a square.
3. Find the value of c that makes x 2  16 x  c a perfect square trinomial. Then write the expression
as a perfect square.
4. Find the value of c that makes x 2  11x  c a perfect square trinomial. Then write the expression
as a perfect square.
5. Build perfect squares and use properties of equality to transform the general conic equation of a
circle into standard form. Then graph the circle.
x 2  24 x  y 2  6 y  137
y 2  24 y  2 x  x 2  120
6. Simplify the following expressions.
7
4 3 15
 
2
9 16 2
8
1 3 1
 
4 4 2
51
7. Use complete sentences to describe what happens to the segments in figure during dilation.
8. What are similar figures? How do similar figures compare with congruent figures?
9. Determine the rate of change or slope in each of the following linear functions.
f (n)  15  3n
f (n)  1  6.5n
f (n)  31n
f (n)  14
f ( n)  7 n  3
10. Determine the median and IQR for the following data set. Create a box plot to represent this
information.
31, 14, 2, 18, 15, 27, 19
11. Write several sentences to describe the different conic sections.
52
We are Similar... Can You Tell?
Choose three integers between -5 and 5: a = b =
c=
Define a coordinate system on the grid below.
Graph the circle with center (a, b) and radius c .
Write your neighbor's numbers here: a =
coordinate system.
b=
c=
and graph their circle on the same
Record your circle's equation in standard form:
Record your neighbor's circle's equation:
*Show that the two circles are similar.
Are all circles similar? Explain your response.
53
Practice Set
Graph the following circles.
1. ( x  1)2  ( y  3)2  25
2. ( x  4)2  ( y  4)2  121
3. ( x  3)2  y 2  25
4. x 2  ( y  2)2  81
Build perfect squares to create an equation of a circle in standard form.
5. x 2  12 x  y 2  18 y  5
6. x 2  24  3  10 y  y 2
7. 7 x 2  28x  7 y 2  14 y  35  0
8. Write several sentences to describe why all circles are similar.
9. Write several sentences to describe the conic sections.
54
10. Complete the following operations.
9 3

2 10
9 3

2 10
9 3

2 10
9 3

2 10
11. Create and label a point of dilation, D. Dilate the image with a scale factor of 1.5.
55
56
Investigate and Report
Your group letter _____
I have an appropriate packet ______
Groups: A. Relationships between measures of arcs and central angles, inscribed angles.
B. Relationships between measures of segments (chords, chord parts, secant segments,
tangent segments)
C. Relationships between measures of angles formed by secants, tangents, radii.
Question to consider throughout the investigation: Do these relationships hold under dilations?
Notes from groups A. Relationships between measures of arcs and central angles, inscribed angles.
57
Practice Set from Group A
1. What is a central angle? Use a diagram to clarify your explanation.
2. Describe the relationship between a central angle and the arc it intercepts.
3. Find the measure of the arc or central angle indicated. Assume that lines which appear to be
diameters are actual diameters.
4. What is an inscribed angle? Use a diagram to clarify your explanation.
5. Describe the relationship between an inscribed angle and the arc it intercepts
58
6. Find the measure of the arc or angle indicated with a question mark.
7. Solve for x.
8. Draw a right triangle and circle such that the right angle is an inscribed angle and the hypotenuse
is a diameter.
59
Notes from groups B. Relationships between measures of segments (chords, chord parts, secant
segments, tangent segments)
60
Practice Set from Group B
Refer to your notes on the previous page!
61
62
Notes from groups C. Relationships between measures of angles formed by secants,
tangents, radii
63
Practice Set from Group C
64
65
15. Determine the missing length of each right triangle.
24
x
x
7
8
25
10
3
x
Label each missing length as rational or irrational. Explain your response.
66
67
Notes from Inscribing and Circumscribing Right Triangles
68
69
Practice Set
1. Describe key information that helped your class find the missing radii in the inscribed circles.
2. Determine the missing side in each triangle.
a. leg length = 6
leg length = 8
hypotenuse length = ?
b. leg length = 7
leg length = ?
hypotenuse = 7 2
3. Determine the value of x in each figure.
55
10
x
2x-17
8
9
4. Use properties of equality to determine the value of the variable.
g
4( x  1)  2 x  7
7 g   11
3
5. Write a function that describes each situation.
a. You started the day with a 450 calorie breakfast. Each minute of your hike burned 2
calories.
b. The trout population in your neighborhood pond was recorded at 36, increasing at a rate
of 2% per year.
70
The Arc de Triomphe roundabouts in Paris, France… on a coordinate system!
Write the equation of the highlighted traffic circle.
Does this traffic circle have any y-intercepts?
Does this traffic circle have any x-intercepts?
If the circle has any intercepts, determine the locations.
Write the equation of a second traffic circle in the image.
If this second traffic circle has any x or y intercepts, determine the locations.
Do these traffic circles intersect? Explain your answer.
Describe how the use of a coordinate system could help in descriptions and calculations of this
traffic scene.
71
Consider the circle ( x  4)2  y 2  9 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
Consider the circle ( x  4)2  ( y  3)2  16 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
72
Consider the circle ( x  5)2  ( y  5)2  4 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
Notes from finding intercepts of circles
73
74
Practice Set
1. Consider the circle ( x  4)2  ( y  2)2  9 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
2. Consider the circle ( x  5)2  ( y  1)2  25 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
3. Consider the circle ( x  1)2  ( y  3)2  16 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
75
4. Consider the circle ( x  6)2  ( y  7)2  121 . Graph.
Does this circle have any y-intercepts?
Does this circle have any x-intercepts?
If the circle has any intercepts, use more than one method to find them.
5. Determine XY.
6. The mB  56 and mAC  82 . Determine the measure of AD .
7. Simplify the following expressions.
8 1
8 1


9 4
9 4
8 1

9 4
8. Write a situation or story that could be described with the following function.
f (n)  30  10.50n
76
Revisiting intersections of a plane and a cone
Use the conic definition and the distance formula to determine the equation of the parabola.
77
How would this direction change the equation?
Examples
78
Given a parabola with vertex at (0,0) and focus at (0,2), sketch the parabola and write the equation.
Given a parabola with vertex at (0,0) and focus at (0,-3), sketch the parabola and write the equation.
Given a parabola with vertex at (0,0) and directrix at y = -5, sketch the parabola and write the
equation.
Given a parabola with vertex at (0,0) and directrix at x = 8, sketch the parabola and write the
equation.
79
Practice Set
Write the equation of the parabola given the following information. Sketch each.
1. Vertex (0,0) and focus (0,4).
2. Vertex (0,0) and focus (2,0).
3. Vertex (0,0) and focus (0,-6).
4. Vertex (0,0) and focus (-7,0).
5. Vertex (0,0) and directrix x = -5.
6. Vertex (0,0) and directrix y = 2.
7. Write the definition of a parabola.
8. What are the reflective properties of a parabola?
9. Given two plans, which would you choose in order to gain the most money in the first ten years?
a. Beginning with $400, the account increases $100 per year.
b. Beginning with $400, the account increases at a rate of 5% per year.
80
Warm-Up Together
Tell whether the parabola opens up, down, left, or right. How can you tell?
x 2  8 y
x 2  12 y
y 2  16 x
y 2  24 x
Graph the equation. Identify the focus and the directrix.
y 2  16 x
x 2  12 y
4x  y2  0
x  2 y2
x2 
9
y
4
y 2  32 x  0
81
Write the equation of the parabola with the given focus, vertex at (0,0).
Focus (0, -1)
Focus (5,0)
A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus. The
reflector is 1 foot deep and 20 feet wide from rim to rim. How far is the receiver from the vertex of
the parabolic reflector?
Write down some ideas on how the equation of our parabola would change if we did not select the
origin (0,0) as the location of the vertex?
82
Name the center of the circle.
x 2  y 2  25
( x  6)2  ( y  1)2  121
In the previous two equations, where did you see the center of the circle?
How might this idea be applied to the equations of parabolas?
Sketch a parabola with vertex at (4,5) and focus at (4,8). Write the equation of the parabola.
Sketch a parabola with vertex at (4,5) and focus at (0,5). Write the equation of the parabola.
Determine the vertex, focus, and directrix of the parabola. Draw a sketch to assist you.
( x  9)2  12( y  2)
83
Practice Set
1. How does the equation of a parabola change if the vertex moves from (0,0) to another location?
2. Write the equation of a parabola with the vertex at the origin and focus at (0, 4.5).
3. Write the equation of a parabola with the vertex at the origin and the directrix at y = 5.25.
4. Write the equation of a parabola with the vertex at (-5,8) and focus at (-15, 8).
5. Write the equation of a parabola with the vertex at (-8,-9) and directrix at x = 0.
6. Find the x and y intercepts of the circle given by the equation ( x  5)2  ( y  11)2  49 .
7. Determine the missing number to build a perfect square trinomial. x 2  14 x  ?
Then write as a perfect square.
8. Build perfect squares to write the equation of the circle in standard form.
x2  4x  y 2  6 y  9  0
84
9. Determine a point of dilation and dilate the figure with a scale factor of 7/2.
10. Use properties of equality to solve for c.
11. Simplify the following expressions.
3 9
3 9


5 10
5 10
9(c  10)  2c  14
3 9

5 10
85
Describe the following equations. Rewrite the expressions in different forms.
( x  4)2  3( y  2)
( y  1)2  12( x  11)
x 2  4( y  9)
Determine the vertex, focus, and directrix and sketch the graph of the parabola.
x 2  4 x  8 y  28
y2  4x  8 y
How does the equation of a parabola compare to the equation of a circle?
86
We are Similar... Can You Tell?
Choose three numbers between -5 and 5: a =
b=
c=
Graph a parabola with vertex (a, b) and a coefficient c (hint: c = 4p in the equation).
Do the same with your neighbor’s choices for a,b,and c.
Record your parabola's equation:
Record your neighbor's parabola's equation:
Show that the two parabolas are similar.
87
Notes from Are We Similar?
88
Practice Set
1. Explain how you know that all parabolas are similar.
2. Compare the equations of parabolas and circles.
3. Find the vertex, focus and directrix of the parabola. Sketch.
x 2  24 y  12
( x  4) 2  12( y  2)
x 2  8 x  2 y  10
x 2  10 x  32 y  1  0
4. Write the equation for the parabola with vertex at (1, 0) and directrix at x = -5.
5. Write the equation for the parabola with focus at (-4,0) and directrix at x = 2.
6. Describe how to slice a cone to get a parabolic cross-section.
89
7. Sketch the circle. Use algebra to find the x and y intercepts.
( x  4)2  ( y  3)2  25
8. Describe how to slice a cone to get a circular cross-section.
9. Describe the effects of dilation on a line segment.
10. Compare linear and exponential functions.
11. Explain which properties are used in the following example of finding the value of x.
8( x  3)  1  2 x  17
8 x  24  1  2 x  17
8 x  2 x  40
6 x  40
20
x
3
90
What is a function?
Which of the following are functions? Make a scatterplot of each.
What is the vertical line test?
Write function equations for as many of the above data sets as you can.
91
Are all parabolas graphs of functions?
Consider the area from this toothpick pattern.
Use multiple representations to describe the area as it changes with each step.
92
Consider the total number of dots in each step of the pattern. Use multiple representations to
describe the total number of dots as it changes with each step.
In general,
93
Tally the total number of segments that can be drawn among n points, no three of which are
collinear. Begin with two points.
94
95
Practice Set
Find the first and second differences for each set of data.
Use these differences to model each of the following data sets with a quadratic formula.
1.
X
0
1
2
3
4
2.
3.
X
0
1
2
3
4
Y
-4
3
14
29
48
5.
X
0
1
2
3
4
Y
.25
.75
3.25
7.75
14.25
7.
X
0
1
2
3
4
X
0
1
2
3
4
Y
3
-4
-15
-30
-49
4.
Y
1
5.5
12
20.5
31
X
0
1
2
3
4
Y
-7
-2
9
26
49
Y
5
3
3
5
9
X
1
2
3
4
5
Y
9
32
69
120
185
X
1
2
3
4
5
Y
-4
-6
-6
-4
0
6.
8.
96
97
98
Notes from Tile Table
99
Consider the total number of toothpicks in the figures below. Use multiple formats to describe the
pattern.
Two students looked at the total number of toothpicks as well. Their thoughts are diagramed below.
In what ways do the following expressions represent the figures below them?
100
Practice Set
Instructions: analyze each table. Classify the pattern as linear or quadratic. Find the rate of change,
and write a function for each.
1.
1
-3
2
-9
3
-15
4
-21
5
-27
6
-33
7
-39
3
30
4
42
5
56
6
72
7
90
8
110
0
-3
1
-7
2
-7
3
-3
4
5
5
17
6
33
2.
3.
101
-2
-3
-1
.5
0
4
1
7.5
2
11
3
14.5
4
18
-3
-28
-2
-11
-1
0
0
5
1
4
2
-3
3
-16
-2
17
-1
11
0
5
1
-1
2
-7
3
-13
4
-9
5.
6.
102
7. In #3, the values in the output column start out decreasing, and then they change to
increasing (you may want to make a graph to verify this). Could that ever happen with a table of
values for a linear pattern? Justify your answer.
8. Find the vertex, focus, and directrix of the parabola. Sketch.
( y  3)2  56 x  20
2 x 2  2 x  24 y  7  0
9. Are all parabolas graphs of functions? Explain your response.
10. Are all parabolas similar? Explain your response.
11. What transformations result in congruent figures?
12. What transformations result in similar figures?
103
In the game “Angry Birds”, little birds are launched at targets. The path of the bird flying through
the air is a parabola. Use the width of a bird as your scale. Find the equation of the parabola.
104
All Angry Birds that are launched move at the same acceleration. The way to select a different
target is by changing the angle of the launch. Choose your next target, select 3 points that will
define the parabola that leads to the target, then find the equation of the parabola.
105
You made it to Lake Powell. It is probably hot and you are waiting around for your parents to take
care of everything, so put on your swimsuit and go jump off the marina (make sure the water is
deep enough or just pretend to jump).
The path of your dive can be modeled by the equation y = -16t2 + 8t
Fill in the table:
Now explain the meaning of the “t” and “y” values.
Not sure? Perhaps a graph would help. Create a graph using the points.
When do you hit the water?
How do you know that?
What is the highest you jump?
At what time are you at the highest point of your jump?
At what time interval are you going up?
At what time interval are you going down?
106
After you have had a chance to anchor the houseboat in deeper water, do the following problems.
Assuming the deck of the houseboat is 10 feet above the surface of the water and you jump from the
deck. Your dive can be modeled by the following equation. y = -16t2 + 6t + 10
Create a table using appropriate values for t.
When did you hit the water?
According to your table, what is the highest point of your jump and at what time were you at the
highest point?
Is this time half-way between the beginning time and the time you hit the water?
Why or why not?
Find the height at t = 3.75
How can you use this information to find the actual time that you reached the highest point of your
dive?
107
Cliff diving at Lake Powell is prohibited; however, if you dived off a cliff that is 30 feet tall, the
dive might be modeled by the equation y = -16t2 +4t + 30. Once again, create a table using
appropriate values for t.
What does t = 0 represent?
What does the height at t = 0 represent?
What does “t” refer to at the time ”y” = 0?
Find the height at t = .125 and use this information to find the maximum height and the time at
which it will occur.
Now suppose the crazy person in your group decides to jump off the 50 foot cliff, write an equation
that might model his/her dive.
Construct a graph of a parabola that opens down.
Now construct several horizontal lines intersecting the graph at several locations.
How can the intersection points of each horizontal line with the parabola be used to find the location
of the maximum value of the parabola?
108
With your group, brainstorm and take notes on the benefits of each of the following quadratic
formats.
( x  7)2  4( y  8)
f (n)  2n2  12n  2
5x 2  10 x  8 y  4  0
( x  5)2  2( y  3)
f ( x)  3x 2  4 x  5
3x 2  8 x  5 y  6  0
109
Practice Set
Write each of the following quadratic equations in a form that is easy to see the coordinates of the
vertex.
Sketch the circle. Use algebra to find the x and y intercepts.
7. ( x  10)2  ( y  1)2  36
8. ( x  7)2  ( y  4)2  81
9. Compare the equations of circles and parabolas.
10. Compare the rates of change of linear, exponential, and quadratic functions.
110
Although the playing surface of a football or soccer field appears to be flat, its surface is actually
shaped like a parabola so that rain runs off to either side. The cross section of a field with synthetic
turf can be modeled by
where x and y are measured in feet. From
this equation, find the width of the field. What is the maximum height of the field’s surface?
Considering the maximum height in the middle of the field, how much higher is the field at the
vertex than at the sidelines (lowest point). Calculate the slope from the vertex to the sideline (slope
of the secant line). Do you think this slope would be detectable by the players on a field?
111
A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y
represents the depth of the duck in meters and x represents the time in seconds.
An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation
for the object's height s at time t seconds after launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in
meters. When does the object strike the ground?
112
Consider the functions below. Rewrite the equations so that you can more easily find the roots. Find the
roots.
a. f ( x)  x 2  40 x  113
b. f ( x)  x 2  6 x  1
c. y  100 x 2  1500 x
113
114
Practice Set
Find the intercepts for the appropriate variable. In each equation, the “y” has already been set to 0.
115
9. Find the vertex, focus, and directrix of the parabola. Sketch.
y 2  36 x  6 y
10. Find the differences and write a function that describes the data.
X
0
1
2
3
4
Y
9
32
69
120
185
11. Determine the missing term to build a perfect square trinomial. Then write each as perfect square.
x 2  36 x  ?
x 2  ? 144
12. Compare the equations of circles and parabolas.
13. Describe the result of a dilation on a rectangle.
14. Simplify the following expressions.
1 3

11 22
1 3

11 22
1 3

11 22
116
Quickly sketch the graphs of the following functions.
1. Graph y = x2 – 4
2. Graph y = x2 + 4
Where are the roots on each graph?
When you look in the mirror what do you see?
a) Is it real?
b) Is it imaginary?
c) What is it?
d) What does the image see when it looks in the mirror?
e) Is it real? Imaginary?
f) What applications does the image have in the real world?

The roots are not real, but there are images that have applications in the real world,
Describe a way to reflect the parabola without real roots in such a way that the image has roots.
Circle the roots on the graph of the parabola or the roots of the image as needed.
117
Refer to functions 1 and 2, and use algebra to solve.
In your group, discuss whether or not your results correspond to the roots you have circled.
What is i?
What are Complex Numbers?
Simplify and write each as a complex number in the form a + bi.
9
7 + 4i
 25
2 – 6i
2
5 + (-3)i
118
The geometry of Complex Numbers
Plot 5 + 2i on an Argand diagram (on the complex plane).
Calculate the distance from 0 + 0i to 5 + 2i on the complex plane.
Modulus
119
Plot -4i on the complex plane. Determine 4i .
Plot, as accurately as you can, the complex number
3  3i . Determine the modulus.
Find the modulus of any complex number, a + bi.
Plot the complex number 3 + 4i on the complex plane. Calculate 3  4i . Give the coordinates of 7 other
points which are the same distance from 0 + 0i. Plot these numbers on the same complex plane.
120
When adding two complex numbers, what is the geometric change?
121
Practice Set
15. Simplify and write the following as complex numbers, a + bi.
49
6  12
73
122
16. What is the geometric effect of adding two complex numbers on the complex plane?
17. How many different complex numbers exist that have modulus = 5? Explain your response.
18. What is i?
19. Write an equation for the conic section.
Circle with center at (-3,1) and radius 2.
Parabola with vertex at (3,-2) and focus at (5,-2).
20. Determine the value of x.
x
x
7
3
110
12
123
What is the geometric effect of multiplying a complex number by a real number?
124
Plot the number 3 + 4i on the complex plane. Determine the modulus.
a. Using the distributive property, multiply i(3 + 4i). Simplify the result and plot on the same complex
plane.
b. Multiply i(result of part a). Simplify the result and plot on the same complex plane.
c. multiply i(result of part b). Simplify the result and plot on the same complex plane.
d. Describe the geometric result of multiplying a complex number by i.
e. Describe each of the following geometrically on the complex plane.
i2
i3
i4
125
Algebra of complex numbers
(1+3i)(7-5i)
(5 + 3i)(5 – 3i)
Can you create an example where the sum of two complex numbers was a real number?
Can you create an example where the product of two complex numbers was purely imaginary? (real part =
0)
126
electrical current equation
V=I•Z
V is voltage, I is current, Z is impedance
Z=V/I
we can also separate the current and voltage using complex notation
Z = V + Ii
Definitions:
Voltage – the difference in electrical charge between two points in a circuit, units are volts
Current – a flow of electric charge, units are amperes or amps
Impedance – the opposition to current flow in AC circuits. In DC voltages, the term resistance is
used. Impedance is simply the measure of how the flow of electrons is resisted. Units are ohms.
Problems
The impedance in one part of a circuit is 4 + 12i ohms. The impedance in another part of the
circuit is 3 – 7i ohms. What is the total impedance in the circuit?
IT = I1 + I2 =
IT =
The current in a circuit is 8+3i amps. The impedance is 1 – 4i ohms. What is the voltage?
I = 8 + 3i amps
Z = 1 – 4i ohms
V=I•Z=I•Z
V=
127
Practice Set
128
129
Consider the following two complex numbers.
-4 +5i
-4 -5i
Graph each on the complex plane and determine the moduli.
Add the two complex numbers together and describe the result.
Multiply the two complex numbers together and describe the result.
Conjugates
What is the conjugate of 3 – 3i?
What is the sum of 3 – 3i and its conjugate?
What is the product of 3 -3i and its conjugate?
130
How could the conjugate help with division of complex numbers?
The voltage in a circuit is 45 + 10i volts. The impedance is 3 + 4i ohms. What is the current?
V = 45 + 10i volts
Z = 3 + 4i ohms
V=I•Z
I=V/Z
131
Practice Set
1. Write the conjugate of the complex number 6 + 2i.
2. Describe the result of adding a complex number and its conjugate together.
3. Describe the result of multiplying a complex number and its conjugate together.
4. Use conjugates and multiply by one to simplify the following:
5. Graph the equation.
( y  4) 2  8( x  1)
( x  1) 2  ( y  2) 2  1
6. Describe how you would determine x and y-intercepts of a graph, given the equation.
132
Deriving a formula
Find the x-intercepts of the following quadratics.
x 2  8 x  15  y
3x 2  18 x  2  y
4 x 2  bx  1  y
ax 2  2 x  1  y
133
y  ax 2  bx  c
The standard form of a quadratic function is given as f (n)  ax 2  bx  c .
Why would we find the roots to this function?
Practice using the quadratic formula to solve. The “y” variable has already been set to zero.
134
Practice Set
1. Find roots using the quadratic formula. The “y” variable has already been set to zero.
2. Determine whether each represents the equation of a parabola or a circle. Then build perfect
squares to help you graph.
x 2  y 2  6 x  4 y  12  0
y 2  2 x  20 y  94  0
135
3. Use properties of equality to find the x-intercepts.
f ( x)  25  7 x
4 x  3.5 y  11
4. Create a histogram. Determine which measure of center and spread would be most appropriate
(mean and standard deviation or median and IQR). Explain your response and find those measures.
5.3, 4.7, 6.0, 3.5, 5.1, 4.4, 8.1, 61
5. Use the circle properties to determine the value of x.
x
x
130
4
3
136
137
Practice Factoring.
138
If g  h  j  0 , what do you know?
How could your work with the Polynomial Puzzler and the Zero Product Property help to solve
quadratic functions?
x2 – 16x + 64 = 0
2x2 + 7x= 15 – y
y+ x2 = 6x
What do you notice about all the roots on this page?
139
Extra practice
140
141
How does the factoring method compare with other methods of finding roots?
When would you choose to use each method? Give an example.
142
Practice Set
Solve by factoring
Simplify the following. Do not write a calculator approximation.
11. 60
12. 140
13. 90
143
Write a function for the following situations.
14. Doris begins her business with a $150 loan. Every gadget she sells, she makes a $3.50 profit.
15. Randall bought two beta fish for his tank. Surprisingly, after three months, Randall had twenty
beta fish in his tank. After another three months, Randall needed room for 200 beta fish.
x
y
0 12
16.
1 20
2 30
3 42
4 56
17. Use complete sentences to describe congruent figures.
18. Use complete sentences to describe similar figures.
19. Determine the distance between the points (-4, 11) and (2, 10).
20. Determine the length of the hypotenuse of a right triangle, if the lengths of the legs are 14 and
9.
144
Extra factoring practice
Describe your comfort level with factoring.
145
Recall the good old days of linear functions…
f ( x)  4  1x
What is the y-intercept or starting amount?
What is the change per step or slope?
Calculate the x-intercept.
Graph the linear function on the graph paper.
f ( x)  6  1x
What is the y-intercept or starting amount?
What is the change per step or slope?
Calculate the x-intercept.
Graph the linear function on the same coordinate system.
146
We will use these linear functions to build a new function by multiplying them together.
y  (4  1x)(6  1x) . Distribute.
What kind of function is this new function? Determine the y-intercept.
Build a perfect square to determine the vertex of the parabola.
Graph the parabola on the same coordinate system as the two linear functions.
What relationship do you see between the graph of the linear functions and the new parabola?
… and the x-intercepts?
… and the y-intercepts?
147
148
Given two roots, could you determine the quadratic function?
Ex. Given x = 3, x = -7, determine a quadratic function.
Ex. Given x = 6 and x = -9, determine a quadratic function.
Ex. Given x = 5 and x = -1, determine a quadratic function.
Ex. Given x = 
2
2
and x = , determine a quadratic function.
5
7
Ex. Given x = 3i and x = -3i, determine a quadratic function.
149
Given factors, what are the roots?
Given roots, what are the factors? The function?
150
Given roots, what are the factors? The function? Sketch the function.
a. 4i and -4i
b. 3i and -3i
c. 2+5i and 2-5i
d. 5+7i and 5-7i
Factor over the complex number system
x 2  16
x 2  10 x  34
151
152
Practice Set
1. For each problem, draw a sketch of the function.
Given factors, what are the roots?
2. Given roots, what are the factors? The function?
153
3. Expand the expression (x + 3)(x – 5i)(x + 5i) in these two ways:
A. [(x + 3)(x – 5i)](x + 5i)
B. (x + 3)[(x – 5i)(x + 5i)]
Compare and contrast the methods.
4. Given roots, what are the factors? The function?
A. 7i and -7i
B. 3 + 9i and 3 – 9i
5. Write the quadratic formula.
6. Use the quadratic formula to solve x 2  10 x  16  0
7. Simplify. (3 + i)(7 – 5i)
154
Alternative notation: rational exponents
Rewrite each of the following (from rational exponents to radicals or vice versa).
1
1
Ex. 8 3 
Ex.
5
b
Ex. m 6 
Ex.
w
1
Ex. 49 2 
3
Ex. 243 5 =
Ex. 25

1
2
2
Ex. 32 5
155
Practice Set
Solve
156
157
Complete your dominoes. Arrange your completed dominoes to ends that correctly match.
158
159
160
161
162
163
Notes from Cutting Corners
164
Compare and sort each set of data.
a.
b.
c.
d.
e.
f.
g. f (n)  13  2n
h. f (n)  2(n  1)2  11
i. f (n)  5 2n
165
How will you decide what model (linear, exponential, regression) fits real data best?
For each of the following, use technology to graph the data. Experiment to determine which
regression model fits best. Write down your chosen model (equation) and explain your choice. Use
your model to respond to the prediction prompt.
A. The table below lists the total estimated numbers of AIDS cases, by year of diagnosis from 1999
to 2003 in the United States (Source: US Dept. of Health and Human Services, Centers for Disease
Control and Prevention, HIV/AIDS Surveillance, 2003.)
Predict the number of cases in the U.S. in the year 2010.
Year AIDS Cases
1999 41,356
2000 41,267
2001 40,833
2002 41,289
2003 43,171
B. As a chambered nautilus grows, its chambers get larger and larger. The relationship between the
volume and consecutive chambers usually follows the same pattern. The volumes v (in cubic
centimeters) of nine consecutive chambers of a chambered nautilus are given in the table. The
chambers are numbered from 0 to 8, with 0 being the first and smallest chamber.
Chamber, n Volume (cm3), v
0
0 .787
1
0.837
2
0.889
3
0.945
4
1.005
5
1.068
6
1.135
7
1.207
8
1.283
166
C. The following data shows the age and average daily energy requirements for male children and
teens.
1 1110
2 1300
5 1800
11 2500
14 2800
17 3000
D. On Tuesday, May 10, 2005, 17 year-old Adi Alifuddin Hussin won the boys’ shot-putt gold
medal for the fourth consecutive year. His winning throw was 16.43 meters. A shot-putter throws a
ball at an inclination of 45° to the horizontal. The following data represent approximate heights for
a ball thrown by a shot-putter as it travels a distance of x meters horizontally.
What would be the height of the ball if it travels 80 meters?
Distance
(m)
7
20
33
47
60
67
Height
(m)
8
15
24
26
24
21
E. The concentration (in milligrams per liter) of a medication in a patient’s blood as time passes is
given by the data in the following table:
Time Concentration
(Hours)
(mg/l)
What is the concentration of medicine in the blood after 1.5 hours have
0
0
passed?
0.5
78.1
1
99.8
2
50.1
2.5
15.6
167
F. The median income of U.S. families for the period 1970-1998.
Predict the median income in 2015.
Year
1970
1980
1985
1990
1995
1998
Income
9867
21,023
27,735
35,353
40,611
46,737
G. At 1821 feet tall, the CN Tower in Toronto, Ontario, is the world’s tallest self-supporting
structure. [Note: This information is taken from College Algebra: A Graphing Approach by Larson,
Hostetler, & Edwards (Third Edition), page 202.]
Suppose you are standing in the observation deck on top of the tower and you drop a penny
from there and watch it fall to the ground. The table below shows the penny’s distance from the
ground after various periods of time (in seconds) have passed. Where is the penny located after
falling for a total of 10.5 seconds?
Time
Distance
(seconds) (feet)
0
1821
2
1757
4
1565
6
1245
8
797
10
221
H. The table below lists the number of Americans (in thousands) who are expected to be over 100
years old for selected years. [Source: US Census Bureau.] How many Americans will be over 100
years old in the year 2008?
Number
Year
(thousands)
1994
50
1996
56
1998
65
2000
75
2002
94
2004
110
Cases found: http://www.algebralab.org/Word/Word.aspx?file=Algebra_QuadraticRegression.xml
168
Population Trends in China
As the new U.S. Ambassador to China, you have been given the country’s population data from
1950 to 1995. You have been asked to prepare and explain a mathematical model that represents
this growth for the State Department. Use multiple formats in your presentation.
The following table shows the population of China from 1950 to 1995.
Year
1950
Population 554.8
in
Millions
1955
609.0
1960
657.5
1965
729.2
1970
830.7
1975
927.8
1980
998.9
1985
1990
1995
1070.0 1155.3 1220.5
169
Your intern forgot to give you the additional data on population trends in China. Does this new data
affect your choice in models? Explain why or why not.
Year
1983
Population 1030.1
in
Millions
1992
1171.7
1997
1236.3
2000
1267.4
2003
1292.3
2005
1307.6
2008
1327.7
170
Practice Set
Granite School District
For each pattern, 1-6, write the function. Use your function to predict the 50th term in each
sequence. (For the picture patterns, a numerical and verbal representation is sufficient).
7. Rewrite the quadratic to determine the vertex. x 2  8 x  10  y
171
8. Simplify.
6i – 3i
9. Simplify.
90
10. Simplify.
i
i2 
i3 
i4 
11. What is the relationship between the measure of a central angle of a circle and its intercepted
arc?
12. What is relationship between the measure of an inscribed angle of a circle and its intercepted
arc?
13. Draw a diagram of a 90 degree inscribed angle. What do you know about its intercepted arc?
14. Describe the effects of dilation upon a line segment.
15. Rewrite as a radical.
3
a7
172
After some hard work, you have earned a season ticket in the student section of the college football
games. And you are part of each half-time show! Each game, you are given a color card, which can be used
to make designs (as shown below).
In honor of the Month of Mathematics, the design will model y  ( x  18) 2 , using a blue and orange color
scheme. Graph the student section design. Y represents the row number and x represents the seat number.
173
Write a rule for row 9 of the design. How does that rule given instructions for each seat color?
Write a rule for row 25 of the design. How does that rule given instructions for each seat color?
174
Graph each of the following.
A. y > -x2 - 6x – 7
B. y > x2 – 3x + 2
175
For each of the following you are given a rule. Create a number line that can describe the rule in greater
detail.
A. x2 + x > 6
B. x2 + x  2
176
Practice Set
Graph each of the following.
177
For each of the following you are given a rule. Create a number line that can describe the rule in greater
detail.
7. x 2  6 x  8  0
8. 4 x 2  16  0
Write whether the following situations could be modeled with linear, exponential, or quadratic functions.
9. A set of data whose first difference changes, but whose second difference is constant.
10. A set of data whose first difference is constant.
11. A set of data multiplied by a common ratio each term.
Answer the questions based on the two-way frequency table given. Show your work for each.
green eyes
not green eyes
female
male
7
4
18
19
12. Of all females, what percent do not have green eyes?
13. Of all green-eyed people, what percent are females?
14. Of all people, what percent are male?
178
Conics: Ellipses
Describe how an ellipse results from the intersection of a plane and the cones.
Describe similarities and differences between an ellipse and a circle.
Describe similarities and differences between a parabola and an ellipse.
Ellipse derives from the Greek term, elleipsis, which means "falling short". Why do you think this
term was used to describe this figure?
179
Notes on ellipses
180
Equation of an ellipse centered at the origin.
Find the vertices and foci of the ellipse and sketch its graph.
x2 y 2

1
100 25
Find the vertices and foci of the ellipse and sketch its graph.
x2 y 2

1
9 64
181
Write the equation of an ellipse with foci at ( 2, 0) and vertices at (5, 0) .
The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest
from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit
with perilune altitude 100km and apolune altitude 314 km (above the moon). Find an equation of
this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus.
182
Practice Set
183
5. Use complete sentences to compare ellipses with circles.
6. Use complete sentences to compare ellipses with parabolas.
7. Use properties of equality to determine the value of x.
14 x  3
6
2
 17  1
5
x
8. Write a function for each of the following data set.
n
f(n)
n
f(n)
0
1
2
3
4
5
3
12
48
192
768
3072
0
1
2
3
4
5
3
4.1
5.2
6.3
7.4
8.5
184
How will the equation change if the center of an ellipse shifts away from (0,0)?
Write the equation of an ellipse with foci at (0,2) and (0,6) and vertices at (0,0) and (0,8).
Find the vertices and foci of the ellipse and sketch its graph.
( x  3)2 ( y  2)2

1
5
100
185
Find the vertices and foci of the ellipse and sketch its graph.
9 x 2  18x  4 y 2  27  0
x2  2 y 2  6x  4 y  7  0
186
Practice Set
Identify the center, vertices, and foci of each.
Graph each equation.
187
Write the standard equation for the ellipses with the given characteristics.
5. Foci (5, 0) and (-5, 0). Vertices (9, 0) and (-9, 0).
6. Vertices (5, 0) and (-5, 0). Co-vertices (0,4) and (0, -4).
7. Choose a center of dilation and dilate the figure with a scale factor of 1.5.
188
Hyperbolas
Describe how a hyperbola results from the intersection of a plane and the cones.
Describe similarities and differences between a hyperbola and a circle.
Describe similarities and differences between a parabola and a hyperbola.
Hyperbola derives from the Greek term which means "over-thrown" or "excessive". Why do you
think this term was used to describe this figure?
What might hyperbolic geometry (non-Euclidean geometry) have to do with hyperbolas?
189
Notes on hyperbolas
190
How does the hyperbola equation compare to other conic equations?
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.
x2 y 2

1
144 25
Write the equation of a hyperbola with foci at (0, 3) and vertices at (0, 1) .
191
How will the equation change if center changes?
Identify the center, vertices, and foci. Then sketch the graph.
( y  1)2 ( x  1) 2

1
9
16
Write the equation of a hyperbola with foci at (1,3) and (7,3) and vertices at (2,3) and (6,3). Sketch
the graph.
192
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.
16 x 2  9 y 2  64 x  90 y  305  0
y 2  x2  4  0
193
Practice Set
194
5. Describe the relationship between the lengths of secant segments in the picture below.
b
c
d
a
6. Using point Q as the center of dilation, dilate the triangle by a scale factor of 2. Then a scale
factor of 3.
Q
195
Conic Sections, Extra Fun
196
Describe triangle ABC, including measurements of side lengths and angles.
Using A as the point of dilation, dilate the triangle by a scale factor of 3/2.
Describe your procedure. Draw and describe the result.
Using A as the point of dilation, dilate the triangle by a scale factor of 2. Describe your procedure.
Draw and describe the result.
197
Draw any triangle with one vertex at the center of the circles.
Using the center as the point of dilation, dilate your triangle by a scale factor of 1/2.
Describe your procedure. Draw and describe the result.
Repeat using a scale factor of 3.
Use one color to highlight all parallel segments.
Use other colors to highlight congruent angles (same color, same measure).
198
Use one color to highlight all parallel segments.
Use other colors to highlight congruent angles (same color, same measure).
Explain your work.
C
D
B
A
What is the relationship between AD and BC?
If two parallel lines are cut by a transversal…
199
Extra Practice
Identify the angle pairs below.
Determine the unknown angle measure.
Solve for x.
200
Eratosthenes estimated the Earth’s circumference by using the fact that the Sun’s rays are parallel.
He chose a day when the Sun shone exactly down a vertical well in Syene at noon. On that day, he
measured the angle the Sun’s rays made with a vertical stick in Alexandria at noon.
Use one color to highlight all parallel segments in the diagram.
Use other colors to highlight congruent angles (same color, same measure).
Explain Eratosthenes method.
1
of a circle. At the time, the distance from Syene to
50
Alexandria was believed to be 575 miles. What was Eratosthenes estimate of the circumference of
the earth?
Eratosthenes determined that m2 
201
Consider the following diagrams of parabolic dishes.
Use one color to highlight all parallel segments in the diagram.
Use other colors to highlight congruent angles (same color, same measure).
Justify your responses.
For each problem, determine the value of x and/or y. Justify your responses.
202
Practice Set
203
20. Simplify the following. Do not give a calculator approximation.
80
21. Simplify the following.
6 21

7 24
8
6 21

7 24
40a 2
6 21

7 24
22. Solve by graphing.
y  x 2  7 x  12
23. Graph the following circle.
( x  5)2  ( y  3)2  36
Use algebra to find any x and y intercepts.
204
For each of the following, color parallel segments, color congruent angles (same measure, same
color). Then find the missing measure.
205
206
207
208
209
Practice Set
7.
210
8.
9. Answer questions based on the following two-way frequency table.
muppet
other
puppet
one-person
15
20
multiple-person
2
7
a. Of all other puppets, what percent are one-person?
b. Of all one-person creatures, what percent are muppets?
c. Of all multiple-person creatures, what percent are other puppets?
10. Solve for x.
10x+6
104
211
Forming logical arguments
Fill out the Sudoku and provide justification as you go.
212
THE DIGIT PLACE GAME
This game is the number version of the commercially available logic game called Master Mind.
OBJECTIVE: The object of the game is to figure out a secret number using as few guesses as possible.
RULES:
One person thinks of a two-digit number. (We’ll use a three-digit number later.) The other
players try to figure out what the secret number is by using some very specific clues. No
digits are repeated within the secret number and no number begins with 0. A player makes a
guess and the person with the secret number responds with two values. The first value (digit)
stands for the number of correct digits in the guess. The second value (place) stands for the
number of correct digits in the guess, which are in the correct place. The players continue to
make guesses/conjectures and receive responses until they know what the secret number is.
YOU SHOULD NOT SAY THE NUMBER; YOU ONLY NEED TO KNOW IT.
PURPOSE:
Remember that our purpose is to give logical arguments to support conjectures, so be
prepared to explain why you think your conjectures are reasonable.
Use the chart provided to determine the missing number. Write a paragraph explaining your reasoning.
1.
Guess
73
34
67
52
Digit Correct
1
0
1
1
Place Correct
0
0
1
1
Digit Correct
0
1
1
0
1
Place Correct
0
1
1
0
0
2.
Guess
95
46
43
12
87
3. Pick a 3-digit number and play the Number Game with a partner.
213
UNO Rules
*Try to use all your cards.
Green 3 is showing. Your hand consists of Blue 3, Yellow 10, Blue 10, Red 3.
Show you can play the Yellow 10.
214
As a group, use acceptable formats to prove that the sum of all the angles in a triangle is 180  .
Helpful diagram:
Helpful hints:
Color parallel segments.
Color congruent angles.
215
Practice Set
Fill out the Sudoku puzzle and provide justification as you go.
Uno proof
Given: Blue 4 is showing. Your hand of cards consists of Green 8, Red 6, Blue 8, Green 4, Red 9,
Yellow 6, Blue 9, Yellow 10.
Prove: You can play the Yellow 10.
Play a game or two of minesweeper with a friend or family member.
216
Paper Fold:
On one side of the paper, label two edges as 8 1/2 inches and two edges as 11 inches. On the
opposite side of the paper, draw a happy face, a tree, or some other simple drawing.
Fold the paper along one diagonal of the rectangle and cut/tear along the diagonal.
Determine the length of the diagonal and record it along the hypotenuse of either or both right
triangles.
Take one of the two right triangles and set it aside. Take the other congruent right triangle and fold
a perpendicular to the hypotenuse that passes through the vertex of the right angle. Cut or tear the
triangle along this fold.
Overlap the three triangles as shown.
Color all congruent angles (same measure, same color). Color all parallel segments.
Determine the lengths of all sides of all three triangles. Write the dimensions on the triangles.
Determine the area of each triangle.
Do the squares of the legs of each right triangle sum to the square of the hypotenuse? Explain why
or why not.
How do the ratios of the sides of the three triangles relate to the ratios of their areas?
Reassemble your paper for reference.
217
Extra Practice Similar Triangles
218
219
Practice Set
220
You will be measuring the lengths of the sides of your triangle and comparing those lengths.
You will then tell the class about your measurements and your comparisons.
Brainstorm about what you need to accomplish these tasks and make our class discussion clear.
Make a sketch of your group triangle. Triangle Number _______
*Note: do not write on your cardboard triangle, please.
Record the measurements of the lengths of sides of your group triangle.
Compare the lengths of sides of your group triangle. Use the notation and methods from the class
discussion.
Comparison 1
Comparison 2
Comparison 3
*Write your group comparisons on three note cards for display.
*Use poster putty to hang your triangle on the front board.
Do not hang your note cards.
221
Class volunteer “sorter” _____________________________
Describe how _____________________ sorted the triangles on the front board.
Send a group member to the board to post your comparison note cards adjacent to your triangle.
Describe the comparison note cards for triangles in the same category.
Color your angle.
Label each side as hypotenuse, adjacent leg, or opposite leg.
Label and set up ratios of sides.
Using the vertex of your angle as the point of dilation, draw a dilated triangle with scale factor 3.
Include lengths on each segment.
Label and set up ratios of sides for the dilated triangle.
222
Create a right triangle and color in the angle of focus. Provide appropriate labels.
Sine: means “pocket”, mistranslation from Sanskrit “chord”
Cosine: “complement” of sine.
Tangent: “to touch”
The ratio comparing the opposite leg to the hypotenuse of a right triangle is called the sine
because...
The ratio comparing the adjacent leg to the hypotenuse of a right triangle is called the
cosine because...
The ratio comparing the opposite leg length to the adjacent leg length of a right triangle is
called the tangent because...
223
Extra Practice
224
Practice Set
225
11. Use algebra to determine any x and y intercepts.
y  14  2 x
( x  2) 2  ( y  3) 2  4
( y  3)  4( x  2) 2
12. Describe how to slice a cone (or double-cone) to get the following cross-sections.
Circle
Parabola
Ellipse
Hyperbola
226
The County building department has given you a scale drawing of the new ramp to be constructed
outside of Farley’s Place.
Color the angle of interest.
Label the sides as hypotenuse, adjacent leg, opposite leg.
Label and set up ratios of sides.
Upon construction at Farley’s Place, the ramp has passed inspection. However, the length of the
ramp surface is 4.7 meters. Determine the other dimensions of the ramp.
227
The calculator screen reports that sin17  .292. Explain how this value could be useful.
Use the information below to draw at least two possible triangles.
cos60  .5
tan 45  1
Angle of elevation.
Angle of depression.
228
Practice Set
229
230
Determine the height of the flag pole in front of school (without measuring the height!).
231
Unit Circle
232
Radian Measure
Convert to radian measure.
120
330
225
Convert from radian measure to degrees.
7
6
4
3

6
233
Draw an angle in standard position on the unit circle. Make it large enough that you can take notes.
Color your angle.
Label your sides as hypotenuse, adjacent leg, opposite leg.
Include any lengths that you know, without taking measurements.
Label and set up ratios of side lengths.
Set up the Pythagorean theorem for this triangle.
Given sin  
4
, find the value of cos .
5
234
Determine the value of  .
235
Reciprocal Definitions
Cosecant
Secant
Cotangent
Variations on the Pythagorean Identity
236
Practice Set
1. Convert from degrees to radian measure.
315
150
2. Convert from radian measure to degrees.
7

4
4
Given each of the following, define the other five trig ratios. Also determine the value of  .
237
4. Write out the Pythagorean Identities.
5. Use algebra to determine the x and y intercepts.
y  5  2 x  10
( x  13) 2  y 2  100
6. Describe the relationship between a central angle of a circle and its intercepted arc. Draw a
diagram.
7. Choose a center of dilation and dilate the figure with a scale factor of 2.
238
On the grids below, and using your ruler, draw two different right triangles. Label each of the
angles A, B, and C so that angle A and B are acute angles, and angle C is 90˚. Measure the sides
and label them as well.
Triangle #1:
Triangle #2:
What relationship do angles A and B have? What name do we give special angles like this?
Find the sine, cosine, and tangent for both triangles for both angles A and B.
What relationship do you notice between trigonometric ratios of the complementary angles?
Compare your results (remember that your triangles may be different than other group members).
Organize your data below.
239
Use your understanding of the relationship between trig ratios of complementary angles to complete
the following problems. (Note that these measurements are not accurate, so using your calculator
will NOT help you!)
1. Given the triangle and trig ratios below, find sin(β) and cos(β).
sin(α) = ⅓
cos(α) = ¼
β
α
2. Suppose that sin(19˚) = 0.2 and cos(19˚) = 0.98. Find sin(71˚) and cos(71˚).
3. Suppose that sin(θ) = 0.35 and cos(θ) = 0.35. Find θ.
Prove that the relationship between trigonometric ratios of the complementary angles holds true for
the triangle below.
A
c
b
B
a
C
Write a statement that generalizes the relationship between the sine and cosine ratios of
complementary angles.
Did you notice any other interesting things about trig ratios that you’d like share?
240
Proving Identities
General strategies
Begin with the more complicated expression
If no other move suggests itself, convert the entire expression into sines and cosines
Combine fractions by getting a common denominator
Examples. Prove the identity.
1. (x – 1)(x + 2) – (x + 1)(x – 2) = 2x
2.
1 1 2 x
 
x 2
2x
3. (cosx)(tanx + sinxcotx) = sinx + cos2x
4.
1  tan 2 x
 sec2 x
2
2
sin x  cos x
241
242
Practice Set
Use your definitions and theorems to prove the identities.
1. cot x  csc x cos x
2.
cot x sin x  cos x
3.
tan x
 sin x
sec x
4.
tan x cos x  sin x
5.
cot x
 cos x
csc x
6.
sin x sec x  tan x
7.
tan x csc x  sec x
8.
sec x (1  cos x )  1  sec x
9.
sin x (1  csc x )  sin x  1
10.
tan x (1  cot x )  1  tan x
11.
cos x (sec x  1)  cos x  1
12.
csc y (sin y  1)  1  csc y
243
13.
cot z (1  tan z )  cot z  1
14.
sin y tan 2 y cot 3 y  cos y
15.
sin2 x sec2 x  sec2 x  1
16.
(1  tan 2 x ) cos 2 x  1
17.
(1  tan y ) 2  sec 2 y  2 tan y
18.
(cos x  sin x ) 2  1  2 sin x cos x
19.
(sin y  cos y )(sin y  cos y )  1  2 cos 2 y
20.
tan 2 x  1
 1  sec 2 x
cot 2 x  1
244
Ignoring the Obvious but Incorrect
Use your unit circle to assist in the following exercises.
1. Let u  180 and v  90 .
Find sin (u + v).
Find sin(u) + sin(v)
Describe your findings.
Test your idea with a few other pairs of angle measures.
2. Let u  0 and v  360 .
Find cos(u + v)
Find cos(u) + cos(v)
Describe your findings.
Test your idea with a few other pairs of angle measures.
3. Find your own values of u and v that will confirm that tan(u  v)  tan(u )  tan(v) .
245
Two right triangles have been arranged such that the two angles in focus, alpha and beta, are
adjacent. Assume only what you have been given on the diagram.
Label the side lengths and determine sin(   ) and cos(   )
1


246
Two right triangles have been arranged such that the two angles in focus, alpha and beta, overlap.
Assume only what you have been given on the diagram.
Label the side lengths and determine sin(   ) and cos(   )
1


-
247
How could we use our identities and definitions to determine the values of tan(α + β) and tan(α - β).
248
Practice Set
Name ______________________________
Rewrite each of the following as a sum or difference of two angles. Use 30 , 45 ,60 ,90 (and their
equivalent radian measures) if possible.
Prove the identities.
249
250
Plot 3 + 4i on an Argand diagram.
Find the modulus of this complex number.
Polar Form of a complex number
251
There and Back Again
Find the polar form of the complex number
1. 3i
2. -2i
3. 2 + 2i
4. 4 – 7i
Write the complex number in rectangular form.
5. 3(cos30  i sin 30)
6. 8(cos 210  i sin 210)
7. 5(cos 60  i sin 60)
8. 5(cos


 i sin )
4
4
252
Practice Set
Name ___________________________
253
254
255
Notes on Shape Statements
256
257
Notes from Card Set A
258
259
260
261
262
Notes for Apple Pi
263
Consider a circle with radius 3. Dilate the circle by scale factors of 2, 3, 5, 7, and 10, using the
center of the circle as the point of dilation.
Create a sketch of these dilations.
Use different representations to describe the area of each circle. Attempt to generalize any patterns.
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
Practice Set
Name ________________________________
Calculate the area of each of the following figures.
281
282
A designer has given you a floor plan of your new yurt home. Determine the area of the design.
(Ignore wall thickness).
283
Notes from Yurt Area
284
Given the scale for the design below, determine the actual area of the floor plan.
285
Determine the area of the regular pentagon.
If figure is dilated with a scale of 3, what will the area of the dilated pentagon be?
286
287
288
289
290
291
Notes from Fill ‘er Up
292
293
294
295
Notes from Popcorn Anyone?
296
297
298
299
300
301
302
303
304
305
Notes from Propane Tanks
306
307
308
309
Notes from Area and Volume
310
Probabilities Involving Area
A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN.
Determine the probability that a spinner lands on section 3?
Dilate the above spinner by a scale factor of 2. Determine the probability that a spinner lands on
section 3?
Determine the probability of the spinner landing in region A.
311