Carnegie Learning Integrated Math I

Carnegie Learning Integrated Math I

North Carolina Math 1 Supplement to Quadratics

Table of Contents

Module 1

Lesson 1-------------------------------------------------------------------------------------- 1 Activity 1.1------------------------------------------------------------------------- 3 Activity 1.2------------------------------------------------------------------------- 6 Activity 1.3------------------------------------------------------------------------- 8 Activity 1.4------------------------------------------------------------------------ 11

Lesson 2------------------------------------------------------------------------------------- 17 Activity 2.1------------------------------------------------------------------------ 20 Activity 2.2------------------------------------------------------------------------ 24 Activity 2.3------------------------------------------------------------------------ 26 Activity 2.4-------------------------------------------------------------------------31

Lesson 3 ------------------------------------------------------------------------------------- 41 Activity 3.1------------------------------------------------------------------------ 43 Activity 3.2------------------------------------------------------------------------ 46 Activity 3.3------------------------------------------------------------------------ 52 Activity 3.4------------------------------------------------------------------------ 56 Activity 3.5------------------------------------------------------------------------ 60

Lesson 4------------------------------------------------------------------------------------- 67 Activity 4.1------------------------------------------------------------------------ 68 Activity 4.2------------------------------------------------------------------------ 72 Activity 4.3------------------------------------------------------------------------ 74 Activity 4.4------------------------------------------------------------------------ 75

Module 2

Lesson 1------------------------------------------------------------------------------------- 83 Activity1.1------------------------------------------------------------------------- 85 Activity 1.2------------------------------------------------------------------------ 88 Activity 1.3------------------------------------------------------------------------ 90 Activity 1.4------------------------------------------------------------------------ 95 Activity 1.5------------------------------------------------------------------------ 97 Activity 1.6---------------------------------------------------------------------- 100

Lesson 2----------------------------------------------------------------------------------- 109 Activity 2.1---------------------------------------------------------------------- 111 Activity 2.2---------------------------------------------------------------------- 116

Lesson 3------------------------------------------------------------------------------------ 123 Activity 3.1----------------------------------------------------------------------- 125 Activity 3.2----------------------------------------------------------------------- 127 Activity 3.3---------------------------------------------------------------------- 129

Lesson 4------------------------------------------------------------------------------------ 135 Activity 4.1----------------------------------------------------------------------- 137 Activity 4.2----------------------------------------------------------------------- 143 Activity 4.3----------------------------------------------------------------------- 146 Activity 4.4----------------------------------------------------------------------- 150 Activity 4.5----------------------------------------------------------------------- 152

1

Up and Down or Down and Up

Exploring Quadratic Functions

Warm Up

Consider f(x) 5 x2 1 3x 1 4. Evaluate the function for each given value.

1. f(1)

2. f(21)

3. f(2)

4. f(22)

Learning Goals

? Write quadratic functions to model contexts. ? Graph quadratic functions using technology. ? Interpret the key features of quadratic functions in

terms of a context. ? Identify the domain and range of quadratic functions

and their contexts.

Key Terms

? parabola ? vertical motion model ? roots

You have used linear functions to model situations with constant change, and you have used exponential functions to model growth and decay situations. What type of real-world situations can be modeled by quadratic functions?

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LESSON 1: Up and Down or Down and Up ? M5-7 1

GETTING STARTED

Squaring It Up

Maddie is using pennies to create a pattern.

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Figure 1

Figure 2

Figure 3

Figure 4

1. Analyze the pattern and explain how to create Figure 5.

2. How many pennies would Maddie need to create Figure 5? Figure 6? Figure 7?

3. Which figure would Maddie create with exactly $4.00 in pennies?

4. Write an equation to determine the number of pennies for any figure number. Define your variables.

5. Describe the function family to which this equation belongs.

M5-8 ? TOPIC 1: Introduction to Quadratic Functions 2

ACTIVITY Using Area to Introduce

1.1 Quadratic Functions

A dog trainer is fencing in an enclosure, represented by the shaded region in the diagram. The trainer will also have two square-shaped storage units on either side of the enclosure to store equipment and other materials. She can make the enclosure and storage units as wide as she wants, but she can't exceed 100 feet in total length.

100 ft

? Carnegie Learning, Inc. Length of the Enclosure (feet)

1. Let s represent a side length, in feet, of one of the storage units.

a. Label the length and width of the enclosure in terms of s.

b. Write the function L(s) to represent the length of the enclosure as a function of side length, s.

c. Sketch and label a graph of the function on the given coordinate plane. Identify any key points.

2. Describe the domain and range of the context and of the function.

Ask

yourself:

To identify key points on the graph, think about the function you are representing. Are there any intercepts? Are there any other points of interest?

y

3. Identify each key characteristic of the graph. Then, interpret the meaning of each in terms of the context.

a. slope

b. y-intercept

c. increasing or decreasing

d. x-intercept

x Side Length of Storage Unit (feet)

LESSON 1: Up and Down or Down and Up ? M5-9 3

The progression of diagrams below shows how the area of the enclosure, A(s), changes as the side length, s, of each square storage unit increases.

4. Write the function A(s) to represent the area of the enclosure as a function of side length, s.

5. Describe how the area of the enclosure changes as the side length increases.

6. Consider the graph of the function, A(s). a. Predict what the graph of the function will look like.

b. Use technology to graph the function A(s). Then sketch the graph and label the axes.

y

Area of Dog Enclosure

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x

7. Describe what all the points on the graph represent.

M5-10 ? TOPIC 1: Introduction to Quadratic Functions 4

The function A(s) that you wrote to model area is a quadratic function. The shape that a quadratic function forms when graphed is called a parabola.

8. Think about the possible areas of the enclosure.

a. Is there a maximum area that the enclosure can contain? Explain your reasoning in terms of the graph and in terms of the context.

Think

about:

Quadratic functions model area because area is measured in square units.

b. Use technology to determine the maximum of A(s). Describe what the x- and y-coordinates of the maximum represent in this context.

c. Determine the dimensions of the enclosure that will provide the maximum area. Show your work and explain your reasoning.

9. Identify the domain and range of the context and of the function.

10. Identify each key characteristic of the graph. Then, interpret the meaning of each in terms of the context.

a. y-intercept

b. increasing and decreasing intervals

c. symmetry

d. x-intercepts

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LESSON 1: Up and Down or Down and Up ? M5-11 5

ACTIVITY Writing and Interpreting a

1.2 Quadratic Function

Suppose that there is a monthly meeting at CIA headquarters for all employees. How many handshakes will it take for every employee at the meeting to shake the hand of every other employee at the meeting once?

1. Use the figures shown to determine the number of handshakes that will occur between 2 employees, 3 employees, and 4 employees.

2 employees

3 employees

4 employees

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Ask

yourself:

Can you tell what shape the graph will be?

2. Draw figures to represent the number of handshakes that occur between 5 employees, 6 employees, and 7 employees and determine the number of handshakes that will occur in each situation.

3. Enter your results in the table.

Number of Employees

2

3

4

5

6

7

n

Number of Handshakes

4. Write a function to represent the number of handshakes given any number of employees. Enter your function in the table.

M5-12 ? TOPIC 1: Introduction to Quadratic Functions 6

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