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Transcript of Web view Chapter 6 ¢â‚¬â€œ Linear Functions. Part A: Graphing...

Chapter 6 –

Linear Functions

Part A: Graphing and Modeling

Unit 6 - Vocabulary

1)

rate

2)

average rate of change

3)

interval

4)

linear function

5)

slope of a line

6)

graph

7)

slope-intercept form of the equation

8)

intercepts (x and y)

Day 1: Average Rate of Change

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Warm-Up

Give an example of a rate in everyday life.

What is the rate of change?

A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.

Model Problem

The table shows the average temperature (°F) for five months in a certain city. Find the rate of change for each time period. During which time period did the temperature increase at the fastest rate?

Independent Variable (x) ______________________ Dependent Variable (y)_________________

From month 2 to month 3

From month 3 to month 5

From month 5 to month 7

From month 7 to month 8

Notes

· Because this is a rate of change, we express the final answer as a fraction or as a statement using the word “per.” Always reduce the fractions.

· When subtracting, make sure you write the “to” number first before the “from” number.

Exercise A

The table shows the balance of a bank account on different days of the month. Find the rate of change during each time interval. During which time interval did the balance decrease at the greatest rate?

Independent Variable ___________________ Dependent Variable __________________

From day 1 to day 6 _________________________________ = ________________

From day 6 to day 16 ________________________________ = ________________

From day ____ to day ____ ______________________________ = ________________

From day ____ to day ____ ______________________________ = ________________

Determining Rate of Change from a Graph

When looking at a graph, we can use the following formula for the average rate of change:

Model Problem B

The graph at right shows the distance that a vehicle travels over time.

Find the average rate of change for each interval.

Hour 0 to Hour 1: Hour 1 to Hour 2:

Hour 2 to Hour 3: Hour 3 to Hour 4:

The average rate of change in this example is measuring the ________ of the vehicle.

Model Problem B continued

1) During which interval of time was the average rate of change the greatest?

The least?

2) How do you know this by looking at the graph?

3) Calculate the average speed of the vehicle from hour 1 to hour 4. Was the vehicle traveling at this speed the whole time? Explain.

Exercise B

Exercise B (continued)

The table shows the number of bikes made by a company for certain years.

a) Plot the points on the graph provided and connect them. Think about:

· Which variable is the independent variable? Which is dependent?

· By what units (1’s, 5’s, 10’s, etc) should you count to fit in all the ordered pairs?

·

b) Using this graph, find the average rate of change for each time period.

c) During which time period did the number of bikes increase at the fastest rate?

From year 1 to year 2

From year 2 to year 5

From year 5 to year 7

From year 7 to year 11

The interval that had the highest average rate of change is between year _________ and year ___________ .

Homework – Day 1

2)

a) Graph the data on the axes below.

b) Calculate the rates of change during each 1-hour interval.

From 0 to 1 hour

From 3 to 4 hours

From 1 to 2 hours

From 4 to 5 hours

From 2 to 3 hours

From 5 to 6 hours

c)

Day 2: Identifying Linear Functions

F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Warm-Up

Use the vertical line test to tell whether the graph of each relation is a function.

1) 2)

3) 4)

What is a Linear Function? How do We Know a Function is Linear?

We already learned that a function is a relation in which each value in the domain (x) is paired with exactly one value in the range (y). The vertical line test works because if a vertical line hits a graph twice, it means that a given value for x is paired with more than one y.

A linear function is a function whose graph forms a straight line. Of the functions that we identified above, we can see that only a few form straight lines. In this lesson, we will learn what makes a given function linear.

Method 1: Use the graph. Perhaps the most obvious way to tell if a function is linear is to use the graph. Remember, first you must decide if it is a function, then if it is, tell whether or not it is linear.

Exercise #1 Which of the graphs in the Warm-Up depicts a linear function?

Method 2: Use the table of values.

Let’s look at some of the graphs you did in the Calculator Exercise and see what makes some linear and others not.

Examples of Linear Functions

The following functions are linear functions. Their tables are listed below. Notice that a constant change in x corresponds to a constant change in y. This is the way you tell that a function is linear from its table of values.

y = 3x + 4

x

y

-3

-5

-2

-2

-1

1

0

4

1

7

2

10

3

13

y =x + 3

x

y

-3

4.5

-2

4

-1

3.5

0

3

1

2.5

2

2

3

1.5

y = 2x

x

y

-3

-6

-2

-4

-1

-2

0

0

1

2

2

4

3

6

These functions are linear functions because _______________________________________________

x

y

-4

5

-3

0

-2

-3

-1

-4

0

-3

1

0

2

5

___________________________________________________________________________________

Examples of Functions that are Not Linear

x

y

-2

-8

-1

-1

0

0

1

1

2

8

y = x2 + 2x – 3 y = x3

x

y

0

3

1

2

2

1

3

0

4

1

5

2

6

3

Explain why these functions are NOT linear:

Exercise Tell whether each set of ordered pairs depicts a linear function. If it does, state the rate of change.

x

y

0

-3

4

0

8

3

12

6

16

9

x

y

3

5

5

4

7

3

9

2

11

1

1) 2)

x

y

-4

13

-2

1

0

-3

2

1

4

13

x

y

0

3

2

7

4

11

8

19

14

31

3) 4)

4) Complete the table below so that the ordered pairs depict a linear function with a constant rate of change equal to

0

2

3

4

10

Slope: A Property of a Linear Function

Recall that the linear function’s rate of change is always the same, or constant. This constant rate of change gives the linear function its shape and is called the slope of the line.

Finding the Slope of a Line

By counting boxes on a graph:

By using two points on a line (x1, y1) and (x2, y2)

Method 1: Counting Boxes

Model Problems

1) Find the slope of the line shown. 2) Find the slope of the line below.

3)

Practice Find the slope of each line.

5) Find the slope of the line that contains (2, 5) and (8, 1).

6) Find the slope of the line that contains (5, –7) and (6, –4).

7) Look back at Questions #1-4. Which lines had a slope that was a positive number? A negative number? Zero?

Positive, Negative, Zero, or No Slope

Positive Slope Negative Slope

As x increases, y increases. As x increases, y decreases.

The line rises from left to right. The line falls from left to right.

Zero Slope No Slope

As x increases, y doesn’t change. If the slope is found, a zero is in

The line is horizontal. the denominator, which is makes the slope und