Unit8% - RCAS · Copyright©!2014 CharlesA.DanaCenterattheUniversityofTexasatAustin,LearningSciencesResearchInstituteattheUniversityofIllinoisatChicago,andAgileMind,Inc.!

Copyright © 2014 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc.

Unit 8 Quadratic functions and equations

In this unit, you will build your understanding of quadratic functions by comparing and contrasting them with linear functions. You will learn how to analyze data to determine whether the data have a linear, quadratic, or other nonlinear relationship. Then you will look at quadratic equations in the context of quadratic functions to help you make connections as you use graphical and algebraic techniques to solve equations. You will learn how to apply the quadratic formula and how to use factoring to solve quadratic equations.

You will continue to use the routines and structures of the course, including partner work and Staying Sharps, to grow and develop as a mathematics learner. You will also reflect on you time in the course and how you have grown as a student.

OUTLINE

Topic 22: Quadratic models and equations

Earlier in the course you learned how linear equations relate to linear functions. You learned a variety of strategies for solving linear equations. Now, you will examine quadratic equations and their connection to quadratic functions. You will learn how to use quadratic functions to model problem situations and how to solve quadratic equations using a variety of methods.

In this topic, you will:

• Determine if a relationship represented by a table, rule, graph, or statement can be represented by a quadratic function

• Learn how to use functions of the form y = ax2 + c to represent some quadratic relationships

• Learn how changes in the parameters a and c for y = ax2 + c affect the graph of the parent quadratic function y = x2

• Solve quadratic equations by graphing and explain the meaning of solutions for given situations

• Simplify square roots algebraically and connect the simplified form to the geometric models for square roots

• Solve quadratic equations using the quadratic formula

• Learn how to determine the number of solutions for a quadratic equation

Topic 23: Polynomial addition and multiplication

In this topic, you will use measurement concepts with which you have a great deal of experience (area, perimeter, and volume) to explore a type of algebraic expression called a polynomial.

In this topic, you will:

• Investigate characteristics of polynomials

• Learn to multiply polynomials using concrete models and the distributive property

• Learn to add and subtract polynomials

Topic 24: Factoring and quadratic equations

Earlier in the course, you learned to solve linear equations by undoing operations and by graphing. In recent topics, you learned how to solve quadratic equations using the quadratic formula and graphing, and you also learned how to multiply polynomials.

In this topic, you will put these ideas together. You will:

• Learn how to reverse the process of multiplication to factor quadratic polynomials

• Use factoring to solve quadratic equations

• Connect solving by factoring to the solution methods you learned previously

360 Unit 8 – Quadratic functions and equations


Topic 22: Quadratic models and equations 361


QUADRATIC MODELS AND EQUATIONS

Lesson 22.1 Quadratic patterns

22.1 OPENER Use the pictorial pattern shown to answer the questions.

Figure 1 Figure 2 Figure 3 Figure 4

1. Draw Figure 4 in the pattern.

2. Create a list of the number of squares in the figure for the first five figures in this pattern.

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5

3. Are the data linear, exponential, or neither? How do you know?

4. Why do you think the numbers in this list are called “square numbers?”

22.1 CORE ACTIVITY 1. Complete this table to show that you understand the relationship between the height of any square array and the number of

painted cube faces it has.

Height in cubes Visual description Written description Process Number of

faces painted

1

A 1-‐by-‐1 array has 1 painted face

1

2

4

3



2. Complete the following table for arrays with heights of 4 and 5 cubes.

Height in cubes Visual description Written description Process Number of

faces painted

4

5

3. Use the process column of your table to write a function rule that expresses the relationship between the height of the square array (n) and the number of cube faces (f) you will paint.

4. You used your graphing calculator to graph the scatterplot of your data and the function rule you developed in question 3 to model that data. Use the graphing calculator image shown here to create a sketch of the scatterplot and function rule graph.

5. How does the graph of the function rule compare with the scatterplot?

6. You expanded the viewing window on your graphing calculator to include the second quadrant. Use the graphing calculator image show here to create a sketch of the graph.



22.1 CONSOLIDATION ACTIVITY 1. Think about the square arrays you built in the last activity. What would the arrays look like if you add one column of cubes

to each square array? Two figures have been sketched for you. Sketch figures for the arrays with heights 3, 4, and 5.

Height of 1 cube Height of 2 cubes Height of 3 cubes Height of 4 cubes Height of 5 cubes

2. Use your sketches to complete the table.

Height in cubes Visual description Written description Process Number of cubes

1

A 1-‐by-‐1 array with 1 more cube is 2 cubes

1 1 + 1 = 2 2

2

A 2-‐by-‐2 array with 2 more cubes is 6 cubes

3

4

5

3. Write a function rule to represent the relationship between the height of the array (h) and the total number of cubes (t) in the array.

4. Use the function rule you wrote to determine the number of cubes needed for an array that has a height of 25.

5. Use the function rule you wrote to find the height of an array that has a total of 110 cubes.



HOMEWORK 22.1 Notes or additional instructions based on whole-‐class discussion of homework assignment:

Martin was thinking about the square array problem from class. He wondered what would happen to the function rule when the length of the array was twice the height of the array. He constructed the following figures:

1. Use your sketches to complete the table.

Height in cubes Visual description Written description Process Number of cubes

1

Two 1-‐by-‐1 arrays 2 (1 1) = 2 2

2

Two 2-‐by-‐2 arrays 2 (2 2) = 8 8

3

4

5

2. Write a function rule to represent the relationship between the height of the array (h) and the total number of cubes (t) in

the array.

3. Use the function rule you wrote to determine the number of cubes needed for an array that has a height of 10.

4. Use the function rule you wrote to find the height of an array that has a total of 98 cubes.



STAYING SHARP 22.1 Practicing

algeb

ra skills & con

cepts

1. Evaluate the expression x – y for the following conditions:

a. If x = 5 and y = 2

b. If x = 5 and y = -‐2

2. Simplify the following expressions:

a. 16

b. − 4

c. 2 9

Prep

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Use the figure shown here to answer questions 3 and 4.

3. Find the areas of the four smaller rectangles. (Write the value for the area inside each rectangle in the diagram.)

4. Explain how you could use the areas of the four smaller rectangles to find the area of the large rectangle.

Review

ing pre-‐algebra ideas

5. The height of a triangle is 12 inches and the base of the triangle is 7 inches. What is the area of the triangle?

Answer with supporting work.

6. The height of a triangle is 6 cm and the area of the triangle is 24 cm2. What is the length of the base of the triangle?

Answer with supporting work.





Lesson 22.2 More quadratic patterns

22.2 OPENER As you saw in the last lesson, some situations are not linear or exponential. Carly and her husband, Rodrigo, have run into one of these situations while remodeling their home.

Carly and Rodrigo are preparing to tile their floors. They want to determine the number of tiles they need for any size square floor. Help them determine the number of tiles they will need for a square room with side length of 3 feet.

1. What is the area of a square room with side length of 3 feet?

2. How many 1 ft x 1 ft tiles would be needed to tile the floor of a square room with side length of 3 feet?

3. What relationships do you notice between the side length of the room and the number of tiles needed to tile the floor?

22.2 CORE ACTIVITY

1. Fill in the table for the area of floor covered and the number of 1 ft × 1 ft sandstone tiles you will need.

Edge length of laid tile (in feet) Area covered (sq ft) Number of 1 ft × 1 ft tiles

1

2

3

4

5

2. Show first differences for the edge length of the laid tile and the number of 1 ft × 1 ft sandstone tiles. What do you notice? 3. a. Is the relationship between the edge length and the

number of 1 ft × 1 ft tiles linear? Defend your answer.

b. Is the relationship exponential? Defend your answer.

4. Carly realizes that a rule exists for her pattern. What rule did she come up to express the relationship between the number of

1 ft × 1 ft sandstone tiles, y, and the edge length, x?



5. The grid shown represents a 10 ft × 10 ft room. Sketch the !!

�

12 ft × !!

�

12 ft natural

wood tiles you would need to cover the room if the edge lengths were 2 ft and 3 ft. Then, fill in the next two rows of the table. Use your ability to recognize patterns to complete the table for edge lengths of 4 ft and 5 ft.

Edge length of laid tile (in feet)

Area covered (sq ft)

Number of natural wood tiles

1 1 4

2

3

6. Compare the data in the tables for the numbers of sandstone tiles and the numbers of natural wood tiles you need to cover the same area. Write a function rule that represents the relationship between the edge length in feet, x, and the number of natural wood tiles, y.

7. The grid shown represents a 10 ft × 10 ft room. Sketch the granite tiles you would need to cover the room if the edge lengths were 4 ft and 6 ft.

Fill in the next two rows of the table. Use your ability to recognize patterns to complete the table for edge lengths of 8 ft and 10 ft.

Edge length of laid tile (in feet)

Area covered (sq ft)

Number of granite tiles

2 4 1

4

6

8. Compare the data in the tables for the numbers of sandstone tiles and the numbers of granite tiles you need to cover the same area. Write a function rule to represent the relationship between the edge length in feet, x, and the number of granite tiles, y.

22.2 REVIEW END-OF-UNIT ASSESSMENT You will now review the Unit 7 end-‐of-‐unit assessment.




1. How can you use a table with equally-‐spaced inputs to determine whether a relationship is linear? Exponential?

2. A quadratic relationship can be identified from a table of x-‐y values if there is a constant second-‐difference among the outputs for equally-‐spaced inputs. Apply the test to determine whether each input-‐output table below represents a quadratic relationship.

a. x y -‐3 13 -‐2 7 -‐1 3 0 1 1 1 2 3 3 7

Quadratic relationship? (circle your answer):

yes / no

Evidence:

b. x y 0 1 1 2 2 4 3 8 4 16 5 32 6 64


yes / no

Evidence:

c. x y 0 1 1 5 2 11 3 19 4 29 5 41 6 55


yes / no

Evidence:

d. x y -‐3 -‐10.5 -‐2 -‐7 -‐1 -‐3.5 0 0 1 3.5 2 7 3 10.5


yes / no

Evidence:

3.

Do you remember the Friendship Club Problem from

earlier in the course? In that problem, you worked to find a relationship between the number of people in

the club and the total number of phone conversations that could take place, assuming that each member had to talk to every other member and that a member can

talk to only one other member at a time. The data for the Friendship Club situation is shown. Show that the

data represent a quadratic relationship.

# of members

# of phone conversations

1 0 2 1 3 3 4 6 5 10 6 15 7 21




algeb

ra skills & con

cepts

1. Martha began with $30 in her wallet and made a series of $5 purchases. She used the function y = 30 – 5x to model the amount of money she has left (y) after x number of purchases. Graph the function.

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-20

-15

-10

-5

5

10

15

20

25

30

2. Use the graph you created in question 1 to answer the following questions.

a. What is the x-‐intercept of the graph?

b. What does the x-‐intercept represent in the context of the problem?

Prep

aring for up

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3. Evaluate the expression (5 + 4) (7 + 1). Sketch an area model to support your solution.

4. Use the distributive property to evaluate the expression (5 + 4) (7 + 1). How is this evaluation method related to the method you used in question 3?

Review


5. What is the area of a square that has a side length of 5 cm? Provide a sketch to support your answer.

6. What is the area of a square that has a perimeter of 12 inches? Provide a sketch to support your answer.



Lesson 22.3 Linear or nonlinear?

22.3 OPENER 1. Complete the three tables below for the functions y = 2x, y = 2x, and y = x2.

A. x y = 2x B. x y = 2x C. x y = x2

0 0 0

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

2. Which of these tables do not represent linear functions? Justify your answers in two different ways.

22.3 CORE ACTIVITY

1. Use your graphing calculator to create the graphs of y = x2, y = 4x2, and y = !!

�

14 x

2. Sketch the graphs below.

Sandstone: y = x2

-15 -10 -5 0 5 10 15

-15

-10

-5

5

10

15

Natural wood: y = 4x2

-15 -10 -5 0 5 10 15

-15

-10

-5

5

10

15

Granite: y = !!

�

14 x

2

-15 -10 -5 0 5 10 15

-15

-10

-5

5

10

15

2. How do the graphs of the rules for the sandstone, natural wood, and granite tiles compare to one another?

3. Observe what happens to the parabola that is the graph of the function y = ax2 as a changes.

a. What happens to the parabola when the a-‐value is greater than 1?

b. What happens to the parabola when the a-‐value is between 0 and 1?

c. What happens to the parabola when the a-‐value is less than 0?



22.3 CONSOLIDATION ACTIVITY

The graph of the parent quadratic function in f(x) = x2 is shown on the right.

-10 -5 0 5 10

-10

-5

5

10

15

1. Use point by point plotting to graph each of the functions on the graph provided.

f (x )= 2x 2

f (x )= 32x 2

f (x )= 52x 2

f (x )= 3x 2

f (x )= 4x 2

-10 -5 0 5 10

-10

-5

5

10

15

2. How do the graphs from question 1 compare to the graph of the parent function f(x) = x2?

3. Use point by point plotting to graph each of the functions on the graph provided.

f (x )= 12x 2

f (x )= 13x 2

f (x )= 23x 2

f (x )= 14x 2

f (x )= 34x 2

-10 -5 0 5 10

-10

-5

5

10

15

4. How do the graphs from question 3 compare to the graph of the parent function f(x) = x2?




1. Without graphing, discuss the similarities and differences between the graphs of the two functions y = 3x2 and y = !!

�

13x2.

2. Use point by point plotting to construct the graphs of the two functions from problem 1. Record any other similarities or differences between the two graphs of the functions.

y = 3x2

y = !!

�

13 x2

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!

3. Sketch a graph of the function y = ax2 where:

i. 0 < a < 1 ii. a < 0 iii. a > 1




algeb

ra skills & con

cepts

1. Simplify each of the following expressions:

a. 36

b. 2+ 16

c. 9 −1

2. Evaluate the expression 2a + b for the following conditions:

d. If a = 3 and b = 1

e. If a = -‐4 and b = 10

f. If a = 5 and b = -‐7

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3. Graph the function y = -‐2x + 6.

4. Explain how you would use the graph in question 5 to find the solution to the equation -‐2x + 6 = 0.

Review


5. Maria built a small flower garden that is 15 inches wide and 3.5 feet long.

a. What is the perimeter of the garden in inches?

b. What is the perimeter of the garden in feet?

6. Maria built another flower garden with dimensions twice the size of the dimensions of her original garden.

a. What are the dimensions of the new garden?

b. How does the perimeter of the larger garden compare to the perimeter of the smaller garden?



Lesson 22.4 More about quadratic functions

22.4 OPENER 1. Make a sketch of what a 6 sq ft porch attached to the original dining room or kitchen might look like.

Remember that the original room is a square with an area of x2.

2. Carly and Rodrigo used the function rule y = x2 to find the number of tiles for the original room. What function rule would help them to find the total number of tiles for the room and the porch combined?

y = +

3. Create a table for both y = x2 and the function rule you wrote in question 2.

x y = x2

(Original room)

y = _________

(Original room and porch combined)

1

2

5

10

4. What do you notice about the values in the table for both y = x2 and the function rule you wrote in Question 2?

22.4 CORE ACTIVITY 1. Use a graphing calculator to graph both y = x2 and your new

function rule from question 2 in the Opener.

2. How will the graph of the original function rule (y = x2) change if Carly and Rodrigo need only four 1 ft sandstone tiles for the

porch?

3. How will the graph of the original function rule (y = x2) change if Carly and Rodrigo need ten 1 ft sandstone tiles for the porch?



4. How will the 13 extra tiles affect Carly’s rule, y = x2 + 6, for the total number of tiles they need to buy? Build a new function rule that considers the 13 free tiles they are getting from Keyoung.

y =

5. Graph all three rules on your graphing calculator and sketch on the grid below. Use the terms y-‐intercept and vertex to describe how the 13 extra 1 ft × 1 ft tiles transform the parabolas for both y = x2 and y = x2 + 6.

6. In the context of the problem situation, what do the points on the graph in quadrants III and IV mean?

7. Observe what happens to the graph of the parabola function, y = x2 + c as c changes.

a. What happens to the parabola when the c-‐value is greater than 0?

b. What happens to the parabola when the c-‐value is less than 0?

c. How are the graphs of y = x2 + 1 and y = x2 – 1 similar to one another? How are they different?



22.4 CONSOLIDATION ACTIVITY A local radio station is sponsoring a basketball-‐shooting contest during an upcoming basketball game. Brian and Jerry decide to enter the contest, and they begin practicing their shots. The two have very different shooting styles.

Brian

Jerry

1. What do you notice about the shapes of the paths that Brian’s and Jerry’s basketball shots create?

2. Describe how the two paths are similar.

3. Describe how the two paths are different.

4. Which type of function represents the path of each basketball shot—linear, quadratic, or exponential? Explain how you know.



5. Complete each statement about the meaning of the variables in the context of the problem situation.

a. The x-‐axis represents . b. The y-‐axis represents .

6. What is a general form of a quadratic function rule that you can use to represent each basketball shot?

7. What are the two values that you can adjust in this function rule to change the shape of the graph?

8. Use your graphing calculator to try different values of a and c until you can match the parabolas.

Then, use the a and c values you found to write a function rule of the form y = ax2 + c for each basketball shot.

a. Function rule for Brian’s shot:

b. Function rule for Jerry’s shot:

9. Find the coordinates of the highest point of each player’s shot. (What do you notice?)

a. Highest point of Brian’s shot:

b. Highest point of Jerry’s shot:

10. How does the highest point on the graph connect to the function rule? Why does this happen?

11. If Brian wants to make his shot from farther away, how must the shape of the ball’s parabolic path change?

12. How must Brian adjust the function rule y = -‐2x2 + 13 to change the shape of his shot in this way?



HOMEWORK 22.4

Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. Without graphing, discuss the similarities and differences between the graphs of the two functions y = x2 + 3 and y = x2 – 3.

2. Graph the two functions from problem 1. Record other similarities or differences between the two graphs of the functions.

y = x2 + 3 y = x2 – 3 ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!

3. Sketch a graph of the function y = x2 + c where:

i. c < 0 ii. c = 0 iii. c > 0




algeb

ra skills & con

cepts

1. Evaluate the expression a +2b for the following

conditions:

a. If a = -‐3 and b = 6

b. If a = 3 and b = 11

c. If a = 18 and b = -‐7

2. Evaluate the expression 3+ a −b for the following

conditions:

a. If a = 4 and b = -‐5

b. If a = 10 and b = 1

c. If a = 4 and b = 4

Prep

aring for up

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3. The area for each of the four smaller rectangles is given in the diagram. Find the two missing dimensions of the rectangle. (Write the missing information in the spaces provided in the diagram)

4. Explain how you found the missing dimensions.

Review


5. Place the following numbers on the number line by drawing and labeling a dot in the location of the number on the number line.

7.2 5 -‐3.3 17 83 − 53

6. Explain how you knew where to place − 53 .



Lesson 22.5 Solving by graphing

22.5 OPENER Sandra’s science class launches a fireworks rocket from the ground. In a table, the students record the height of the rocket in feet above the ground at specific times after the launch.

Time in seconds 0 1 2 3 4 5 6 7

Height in feet 0 144 256 336 384 400 384 336

1. At time , the height of the rocket is 0 feet. What does this mean?

2. Based on the data in the table, what type of function (for example, linear, exponential, or quadratic) can be used to model the data? Justify your answer.

3. Based on the data in the table, what is the maximum height

the rocket appears to reach?

4. Based on the pattern in the table, what will the height of the rocket be at 9 seconds?

22.5 CORE ACTIVITY 1. Complete the table to show the relationship between elapsed time and balloon height. Plot the data on the graph.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-10

-5

5

10

15

20

25

30

35

40

45

50

55

60

Time (seconds)

Height (feet)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Heigh

t (fee

t)

Time (seconds)

2. Write the function that Vincent and Bobby use to model the balloon’s height. Use your graphing calculator to graph this function, and sketch its graph on the same grid you used to plot the data in question 1.

3. Based on the data in the table, what is an appropriate domain for the balloon launch situation? What is an appropriate range?

4. What equation models this situation when the height is 0 feet? 5. When will the balloon hit the ground? Justify your answer.




1. The graph of quadratic function is a curve called a parabola.

a. Parabolas have a line of symmetry. Draw a line of symmetry on each of the following graphs. i.

ii.

-6 -5 -4 -3 -2 -1 1 2 3

-5-4-3-2-1

123456789

10

x

y

b. Each of the following graphs shows part of a parabola. Use the fact that parabolas have symmetry to sketch in the missing part of the parabola.

i.

-6 -5 -4 -3 -2 -1 1 2 3

-5-4-3-2-1

123456789

10

x

y

ii.

2. These tables for quadratic relationships are not complete. Use what you know about quadratic functions to complete the missing entries in the tables.

a.

b. x y -‐4 14 -‐3 6 -‐2 0 -‐1 -‐4 0 -‐6 1 -‐6 2 -‐4 3 4 5



3. Estimate x-‐intercepts for each of the following graphs. For the graphs on graphing calculator screens, assume that each tick mark represents one unit.

a.

-6 -5 -4 -3 -2 -1 1 2 3

-5-4-3-2-1

123456789

10

x

y

b.

c.

d.

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

4. Use the tables to find the x-‐intercepts for the given functional relationships. a.

b. x y

-‐8 -‐22

-‐7 -‐10

-‐6 0

-‐5 8

-‐4 14

-‐3 18

-‐2 20

-‐1 20

0 18

1 14

2 8

3 0

4 -‐10

5 -‐22



5. a. A graph for the quadratic function y x2 9= − is shown.

b. A table for the quadratic function y x2 1= − is shown.

Use the graph to solve the following quadratic equation:

x2 9 0− = .

Solutions: _________________

Use the table to solve the following quadratic equation:

x2 1 0− = .

Solutions: _________________




1. The following graph shows part of a parabola.

a. Use the fact that parabolas have symmetry to sketch in the missing part of the parabola.

b. Sketch in the line of symmetry for the parabola.

c. Estimate the x-‐intercepts. Then mark them on the graph. x-‐intercepts: _______________

d. Find the y-‐intercept. Then mark it on the graph. y-‐intercept: ________

e. Complete the table for the parabola.

X -‐2 -‐1 0 1 2 3 4 5

y

2. Below is a graph for the quadratic function y = –x2 + 4.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-9-8-7-6-5-4-3-2-1

1234

x

y

Shown is a table for the quadratic function y = x2 – 9.

Use the graph to solve the quadratic equation x2 4 0− + = .

Solutions: _________________

Use the table to solve the quadratic equation x2 9 0− = .

Solutions: _________________

3. Jocelyn’s sister tells her that the table of values shown here is from a quadratic function. Jocelyn knows this means that the graph of the function is a parabola and that there is a line of symmetry at an x-‐value. Help her to find the x-‐value at the line of symmetry. Explain your reasoning.

x -‐4 -‐3 -‐2 -‐1 0 1 2 y 10 7 6 7 10 15 22




algeb

ra skills & con

cepts

1. If the length of a side of a square is 10 inches, what is the area of the square?

Answer with supporting work:

2. If the area of a square is 64 square centimeters, what is the length of the side of the square?


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3. The expression b− is read as “the opposite of b .”

a. Evaluate b− if b 2= .

b. Evaluate b− if b 10= − .

c. Evaluate b− if b 0= .

4. Evaluate the expression ac4− for the following conditions:

a. If a 2= and c 3= −

b. If a 4= − and c 2= −

c. If a 1= − and c 5=

Review


5. The Pythagorean Theorem states that for a right triangle with legs of length a and b and hypotenuse of length c, .

Use the Pythagorean Theorem to find the length of the missing side in the right triangle shown here.


6. The lengths of the sides of!ΔABC are proportional to the lengths of the sides of !ΔDEF . Use the information

in the diagrams to find the lengths of !DE and !EF .




Lesson 22.6 Approximating solutions

22.6 OPENER Place the following values on the number line. You will need to estimate the locations for some values.

8 8 23 3 9 4 2

22.6 CORE ACTIVITY Bobby and Joseph drop a water balloon from a height of 10 meters.

1. What rule might represent the balloon’s height above the ground, h, in meters as a function of time, t, in seconds?

2. On your calculator, graph the function rule you wrote in question 1. Sketch the graph on the calculator screen image shown here. What does the graph tell you about the water balloon drop? Use your graph to estimate how long it will take the balloon to hit the ground.

3. Write an equation that can be solved to find when the water balloon will hit the ground.

4. Use your calculator to find a solution to the equation you wrote in question 3 that makes sense in the context of the Balloon Drop Problem.

5. In your own words, describe what each mathematical term means.

x-‐intercept solution zero

6. What does the point (1.429,0) means in the balloon drop situation?



22.6 CONSOLIDATION ACTIVITY Sandra’s science class launches a fireworks rocket from the ground. In a table, the students record the height of the rocket in feet above the ground at specific times after the launch.

1. Create a plot for the data, using the coordinate grid provided.

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-200

-150

-100

-50

50

100

150

200

250

300

350

400

Time in seconds Height in feet

0 0

1 144

2 256

3 336

4 384

5 400

6 384

7 336

8

9

10

Heigh

t (in

feet)

Time (in seconds)

2. The data and plot represent a quadratic relationship. How can you use the data in the table to show that the relationship is quadratic?

3. What are some features of quadratic graphs? (Recall that the shape of a quadratic graph is called a parabola.)

4. Use the features of a quadratic relationship, including the symmetry of the data/graph, to complete the table.

5. Add the new data points to the graph in question 1.

6. What are the x-‐intercepts of the graph? How can you identify the x-‐intercepts from the graph?

7. How can you identify the x-‐intercepts from the table? 8. What do the x-‐intercepts mean in the context of the rocket problem situation?

9. Sketch the path that the rocket takes. Assume that the rocket is shot straight up.

10. Is the rocket’s path the same as or different from a graph of height vs. time for the rocket? Explain.




1. Courtney’s model rocket is shot straight up into the air from a launcher. Some data for the rocket are shown in the table. Plot the data on the grid provided.

Time in seconds

0 1 2 3 4

Height in feet

0 80 128 144 128

Heigh

t (fee

t)

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

-200

-175

-150

-125

-100

-75

-50

-25

25

50

75

100

125

150

Time (seconds)

2. The function that describes the height of the rocket at a given time is quadratic, so its graph is a parabola. Use the fact that a

parabola has symmetry to plot the height until the rocket lands back on the ground. Also complete the following table and answer the questions.

Time in seconds 0 1 2 3 4 5 6

Height in feet 0 80 128 144 128

a. What is the maximum (highest) height that the rocket reaches? At what time does this happen?

b. What are the x-‐intercepts of the parabola?

c. What do the x-‐intercepts mean in terms of the height of the rocket?

d. The graph you created shows the height of the rocket as a function of time. Does it also show the path the rocket takes while it is in the air? Explain.



3. Griffin also shoots his model rocket straight up into the air. Data showing the height of his rocket (in feet) as a function of time (in seconds) are shown in the following table.

Time in seconds 0 2 4 6 8 10 12

Height in feet 0 320 512 576 512 320 0

a. Whose rocket went higher, Courtney’s or Griffin’s? How

much higher did it go?

b. Whose rocket took longer to land? How much longer did it take to land?

4. An equation that models the height of Griffin’s rocket, h(t), as a function of time, t, is h(t) = –16t2 + 192t. What is the value of h(2)? What does h(2) represent in terms of the rocket?

5. Courtney and Griffin’s physics teacher told them that the height of a rocket as a function of time will always be quadratic when fired from a launcher as in this experiment. Griffin shoots his rocket from the ground another time and collects the data shown in the table. He knows that it took 8 seconds for the rocket to land. Looking at the data, however, he realizes that the device that measures the height of the rocket malfunctioned. How does he know that just by looking at the data?

Time in seconds 0 1 2 3 4 5 6 7 8

Height in feet 0 112 192 240 256 240 105 90 0




algeb

ra skills & con

cepts

1. The area of the rectangle shown can be written as .

Use the distributive property to write the expression in another way.

2. Write the area of the rectangle shown as a product of the length and width. Then use the distributive property to rewrite the expression in another way.

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3. Explain why it is not possible to simplify each of the following square roots.

a.

b.

4. Simplify each of the following expressions.

a.

b. !!

�

3 + 32

c. !!

�

5 + 16

Review

ing pre-‐algebra skills

5. In a particular blueprint drawing, inch represents 1

foot. If the length of a bedroom is 3 inches on the

blueprint, what is the actual length of the bedroom?


6. In a particular blueprint drawing, inch represents 1

foot. The actual width of the living room will be 22 feet. What is the width, in inches, of the living room on the blueprint drawing?






Lesson 22.7 The geometry and algebra of square roots

22.7 OPENER Consider the quadratic equation x2 = 36.

1. Solve the equation by graphing. Use your graphing calculator. (Remember to use a viewing rectangle that will allow you to see the solutions and the parabola.) Sketch your graph below and indicate the solutions to the equation.

2. How do your solutions connect to what you have learned about square roots?

22.7 CORE ACTIVITY

1. Solve using square roots. Then check your answer by creating a graph on the graphing calculator.

2. Each of these quadratic equations has two solutions. Use square roots to represent the exact value for each solution.

a. b.

3. Each of these quadratic equations has two solutions. Estimate the solutions by stating between which two integers each solution lies.

a. b.

4. Now, use your calculator to find approximate solutions to three decimal places for the following quadratic equations.

a. b.



The Pythagorean Theorem states that for a right triangle with legs of length a and b and hypotenuse of length c, a2 + b2 = c2.

5.

Use the Pythagorean Theorem to find the length of the missing side in the triangle shown.

A Pythagorean triple is a set of three positive whole numbers, a, b, and c that are the lengths of the sides of a right triangle. The right triangle in question 5 is an example of a Pythagorean triple because the lengths of its sides are all positive, whole numbers.

6. One way to find other Pythagorean triples from a known Pythagorean triple is to “scale up” (by a whole number) the lengths of the sides of a right triangle like this 5-‐12-‐13 right triangle. Write in the lengths of the sides of the other two right triangles shown.

5-‐12-‐13 right triangle:

5-‐12-‐13 right triangle scaled up by a factor of 2:

5-‐12-‐13 right triangle scaled up by a factor 3:

7. Use the Pythagorean Theorem to find the length of the diagonal of the square.



8. As you watch the animations, fill in the following table.

Side length

of square

Length of

diagonal

Simplified

form

1

2

3

4

9. Justify each step used to simplify the following radical.

10. Write an algebraic equation that is an example of the following statement: The square root of a product of two positive real numbers is the product of their square roots.

11. Use algebra to show that the following statement is true: ! 32 = 4 2 .

12. Simplify the following radical expressions. Show your work to support your answer.

a. ! 12

b. ! 48

c. ! 180 d.

! 75

e.

! 500 f.

!!

�

150



13. Simplify the following radical expressions. Show your work to support your answers.

22.7 CONSOLIDATION ACTIVITY In this topic, you have looked at solving quadratic equations by graphing. You have also looked at how you can use square roots to solve quadratic equations.

In this activity, you will work with your partner to match quadratic equations with their graphs and their solutions.

Objective: Create sets of “matching” cards. “Matching” is defined as different representations of the same relationship. Each set will have an equation card, a graph card, and a solution card.

Materials: Your teacher will give you and your partner pages with cards on them to cut out. There are eight equations cards (labeled A-‐H), eight graph cards (labeled I-‐P), and eight solutions cards (labeled Q-‐X).

Instructions: Work with your partner to find a set of matching cards. When you both agree on a set, tape together the cards that form the set. So that you can easily check you answers, tape each set with the equation card on the left, the graph card in the middle, and the solutions card on the right as shown here.

Equation card

x2 – 36 = 0

Graph card

Solutions card

x = -6 x = 6




1. Between which two consecutive integers is each square root?

a.

b. c.

2. Simplify the following square roots if possible. Write your answers in simplest form. Show your work.

a.

b. c.

d.

e.

3. Matt and Karen are working on their algebra homework when they come across the following problem:

“A particular square has an area of 50 square inches. Approximately how long is each side of the square? Give your answer to the nearest whole number. Explain your reasoning.”

Matt thinks the length of each side is about 7 inches and Karen thinks the length is about 8 inches.

a. Why did they choose these two numbers for their approximations? Explain.

b. Whose answer do you think is closer to the true length of the side? Why?



4. Complete the Math Journal. Consider the process of simplifying square roots. Sometimes it is possible to do so, sometimes it is not possible.

Possible to simplify?

(Yes or no) Explain why it is possible (or impossible) to simplify this square root.




algeb

ra skills & con

cepts

1. The value of is between the whole numbers 2 and 3. Explain why.

2. Between what two consecutive whole numbers is ?

Answer with supporting explanation:

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3. Simplify the following expressions:

a.

b.

4. Evaluate the following expressions if

!a = 1, b = −4, c = 2 :

a. !b2 − 4ac b.

! b2 − 4ac

Review


5. Use the Pythagorean Theorem to find the length of the diagonal of the square.

!a2 + b2 = c2

6. Place in order from least to greatest.






Lesson 22.8 Using the quadratic formula

22.8 OPENER Consider the quadratic equation 2x2 + 3x – 3 = 0. What happens if you try to solve the equation by graphing?

These calculator screens show the results of finding the x-‐intercepts of the graph using technology.

Do you think the x-‐intercepts shown represent exact solutions or approximate solutions? Explain.

22.8 CORE ACTIVITY 1. If ax2 + bx + c = 0, a ≠ 0, then x = _____________________________________.

2. Use the quadratic formula to solve the equation 2x2 + 3x – 3 = 0.

3. How does your solution in question 2 compare to the solution shown in the Opener?

4. Use the quadratic formula to solve the following equations.

a. 3x2 – x – 5 = 0 b. – 6 – 2x + x2= 0




The quadratic formula

For ax2 + bx + c = 0, the value of x is given by.

1. Use the quadratic formula to solve each of the following quadratic equations.

a. x2 + x – 2 = 0

b. x2 – 5x + 6 = 0 c. x2 – 6x = 0

d. x2 + 4x + 4 = 0

e. 2x2 – x – 6 = 0 f. 5x – 8 + 3x2 = 0

2. Randy, Sandy, and Andy are using the quadratic formula to solve the quadratic equation x2 – 4x + 3 = 0. They each make a

mistake when applying the formula. Find the mistake in each of the solutions shown. Then re-‐work the problem to find the correct solution.

Solution Explanation of mistake Corrected solution

a. Randy

b. Sandy

No solutions because you cannot take the square root of a negative number

c. Andy




1. Complete the Math Journal. The quadratic formula tells us that for ax2 + bx + c = 0, the value of x is given by .

When you use the quadratic formula to solve a quadratic equation, the answers you get are values of x that make the equation true.

Question Answer Sketch a graph to support your answer What do the values of x that result from applying the quadratic formula represent in terms of the graph of the corresponding parabola?

2. Use the quadratic formula to solve each of the following quadratic equations.

a. x2 + 3x = 0

b. x2 + 2x – 8 = 0

c. x2 + 11x + 10c = 0

d. 3x2 + 7x + 2 = 0

3. Mallory is trying to use the quadratic formula to find where the parabola defined by y = x2 – 2x + 8 crosses the x-‐axis. Her

friend, Mark, graphs the parabola on his calculator and sees that the x-‐intercepts occur at and . Mallory’s answers do not agree with Mark’s.

Here is Mallory’s work:

; no solutions because you cannot take the square root of a negative number

a. Explain what Mallory did incorrectly.

b. Correct Mallory’s work. Be sure you get the correct answers: and .




algeb

ra skills & con

cepts

1. Evaluate the following expression for

, , and . Show your work.

−b + b2 − 4ac

2. Evaluate the following expression for

, , and . Show your work.

−b + b2 − 4ac

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3. Does the graph of the following linear function have an x-‐intercept? If it does, what is it?


4. Does the graph of the following linear function have an x-‐intercept? If it does, what is it?


Review


5. Betsy’s grandmother estimates that she will need 900 yards of yarn to make a blanket for her. How many feet of yarn is this? Answer with supporting work:

6. One kilometer is about of a mile. If the distance

from Detroit to Cleveland is 170 miles, what is the approximate distance in kilometers? Answer with supporting work:



Lesson 22.9 How many solutions?

22.9 OPENER For each quadratic equation in the table, a graph is shown along with a solution to the equation using the quadratic formula. Study the examples in the table. (Note: Each tick mark on the x-‐axis and y-‐axis of the graph represents one unit.)

Equation Graph Solution

x2 + x – 6 = 0

x2 + x + 2 = 0

No real solutions

What conjectures can you make about the connection between the graph of a quadratic equation and the number of (real) solutions the quadratic equation has?

22.9 CORE ACTIVITY 1. Circle the discriminant in the quadratic formula.

!!x =

−b ± b2 − 4ac2a



2. For each quadratic equation, find the value of the discriminant and determine the number of solutions. Then, use the quadratic formula to solve the equation.

a.

b.

3. Fill in the following table.

If the discriminant is… then the quadratic equation has….

positive

zero

negative

22.9 ONLINE ASSESSMENT Today you will take an online assessment.



HOMEWORK 22.9

Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. Complete the Math Journal. Consider solving quadratic equations (in the form ) using the quadratic formula,

. The value of the discriminant (the expression b2 – 4ac) provides a clue about the number of real solutions

that you will get when you solve the equation.

Value of the discriminant (b2−4ac)

What does this tell you about the number of real solutions you will get for the equation?

Sketch of possible graph for this situation showing number of x-‐intercepts

Case 1: When b2−4ac > 0

Case 2: When b2−4ac = 0

Case 3: When b2−4ac < 0

2. Use the quadratic formula to solve each quadratic equation. In cases that involve negative square roots, write “no real solutions” for the answer. For equations that have real solutions, express your answers as exact answers. Record your work and answers on notebook paper.

a. b. x2 + x – 4 = 0 c.

d. e. f.




algeb

ra skills & con

cepts

1. Recall that some square roots can be simplified by finding “perfect square” factors greater than 1 (such as 4, 9, 16, 25,…).

Example:

If possible, simplify the following square roots:

a.

b.

c.

2. Simplify each expression.

a.

b.

c.

d.

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3. Find the areas of the four smaller rectangles. (Write the value for the area inside each rectangle on the diagram.)

4.

Write an equation that states the area of the large rectangle as the sum of the areas of the four smaller rectangles. Then state the value of the area of the large rectangle.

Review


5. The Pythagorean Theorem states that for a right triangle with legs of length a and b and hypotenuse of length c, .

Use the Pythagorean Theorem to find the length of the missing side in the right triangle shown here.


6. This diagram shows a smaller square inscribed within a larger square. Find the area of the smaller square.


Documents

Unit8% - RCAS · Copyright©!2014 CharlesA.DanaCenterattheUniversityofTexasatAustin,LearningSciencesResearchInstituteattheUniversityofIllinoisatChicago,andAgileMind,Inc.!