5-6 slope and rates of change warm up warm up lesson presentation lesson presentation problem of the...

5-6 Slope and Rates of Change

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

5-6 Slope and Rates of Change

Warm UpGraph the linear function.

y = 2x + 4

5-6 Slope and Rates of Change

Problem of the Day

What two 3-digit numbers have a product of 19,019?

133 and 143

5-6 Slope and Rates of Change

Learn to determine the slope of a line and to recognize constant and variable rates of change.

5-6 Slope and Rates of Change

Vocabulary

slope

rate of change

5-6 Slope and Rates of Change

The slope of a line is a measure of its steepness and is the ratio of rise to run:

y

If a line rises from left to right, its slope is positive. If a line falls from left to right, its slope negative.

x

Run

Rise

5-6 Slope and Rates of Change

Tell whether the slope is positive or negative. Then find the slope.

Additional Example 1A: Identifying the Slope of the Line

The line rises from left to right.

The slope is positive.

5-6 Slope and Rates of Change

Tell whether the slope is positive or negative. Then find the slope.

Additional Example 1A Continued

The rise is 3. The run is 3.

slope = riserun

= 33

= 1

3

3

5-6 Slope and Rates of Change

Tell whether the slope is positive or negative. Then find the slope.

Additional Example 1B: Identifying the Slope of the Line

The line falls from right to left.

The slope is negative.

0 2

2

–2–2

y

x

5-6 Slope and Rates of Change

Tell whether the slope is positive or negative. Then find the slope.

Additional Example 1B Continued

The rise is 2. The run is -3.

slope = riserun

= 2 -3

2

-3

0 2

2

–2–2

y

x

5-6 Slope and Rates of Change

Check It Out: Example 1A

The line does not point upward or downward so it is not positive or negative.

Tell whether the slope is positive or negative. Then find the slope.

5-6 Slope and Rates of Change

Check It Out: Example 1A Continued

The rise is 0. The run is 2.

slope = riserun

= 02

= 0

M(1, –1) N(3, –1)

2

Tell whether the slope is positive or negative. Then find the slope.

5-6 Slope and Rates of Change

Check It Out: Example 1B

(0, –4)

(–2, 4)

8

–2

The rise is 8. The run is –2.

slope = riserun

= 8–2 = –4

Tell whether the slope is positive or negative. Then find the slope.

The line falls from left to right.

The slope is negative.

5-6 Slope and Rates of Change

You can graph a line if you know its slope and one of its points.

5-6 Slope and Rates of Change

Use the slope and the point (1, –1) to graph the line.

Additional Example 2A: Using Slope and a Point to Graph a Line

2 1

From point (1, 1) move 2units down and 1 unitright, or move 2 units up and 1 unit left. Mark the point where you end up, and draw a line through the two points.

y

x–4

–4

4

4

0 2

2

–2–2

= orriserun

-2 1

2-1

●

●

5-6 Slope and Rates of Change

You can write an integer as a fraction by putting the integer in the numerator of the fraction and a 1 in the denominator.

Remember!

5-6 Slope and Rates of Change

Use the slope and the point (–1, –1) to graph the line.

Additional Example 2B: Using Slope and a Point to Graph a Line

1 2

From point (–1, –1) move 1unit up and 2 units right. Mark the point where you end up, and draw a line through the two points.

y

x–4

–4

4

4

0 2

2

–2–2

=riserun

1 2

●

5-6 Slope and Rates of Change

Use the slope – and the point (2, 0) to graph the line.

Check It Out: Example 2A

2 3

From point (2, 0) move 2units down and 3 unitsright, or move 2 units up and 3 unit left. Mark the point where you end up, and draw a line through the two points.

y

x–4

–4

4

4

0 2

2

–2–2

= orriserun

-2 3

2-3

●

●

5-6 Slope and Rates of Change

Use the slope and the point (–2, 0) to graph the line.

Check It Out: Example 2B

1 4

From point (–2, 0) move 1unit up and 4 units right. Mark the point where you end up, and draw a line through the two points.

y

x–4

–4

4

4

0 2

2

–2–2

=riserun

1 4

●

5-6 Slope and Rates of Change

The ratio of two quantities that change, such as slope, is a rate of change.

A constant rate of change describes changes of the same amount during equal intervals.

A variable rate of change describes changes of a different amount during equal intervals.

The graph of a constant rate of change is a line, and the graph of a variable rate of change is not a line.

5-6 Slope and Rates of Change

Tell whether each graph shows a constant or variable rate of change.

A. B.

Additional Example 3: Identifying Rates of Change in Graphs

The graph is nonlinear,so the rate of change is variable.

The graph is linear, so the rate of change isconstant.

5-6 Slope and Rates of Change

Check It Out: Example 3

Tell whether each graph shows a constant or variable rate of change.

A. B.

The graph is nonlinear,so the rate of change is variable.

The graph is linear, so the rate of change isconstant.

y

x–4

–4

4

4

0 2

2

–2–2

y

x–4

–4

4

4

0 2

2

–2–2

5-6 Slope and Rates of Change

The graph shows the distance a monarch butterfly travels overtime. Tell whether the graph shows a constant or variable rate of change. Then find how fast the butterfly is traveling.

Additional Example 4: Using Rate of Change to Solve Problems

5-6 Slope and Rates of Change

Additional Example 4 Continued

The graph is a line, so the butterfly is traveling at a constant rate of speed.

The amount of distance is the rise, and the amount of time is the run. You can find the speed by finding the slope.

slope (speed) = rise (distance)run (time) = 20 miles

1 hour

The butterfly travels at a rate of 20 miles per hour.

5-6 Slope and Rates of Change

Check It Out: Example 4The graph shows the distance a jogger travels over time. Is he traveling at a constant or variable rate. How fast is he traveling?

7 14 21 28 35Time (min)

1

2

3

4

5

6

Dis

tan

ce (

mi)

7 14 21 28 35Time (min)

1

2

3

4

5

6

Dis

tan

ce (

mi)

1

7

7

1

5-6 Slope and Rates of Change

Check It Out: Example 4 Continued

The graph is a line, so the jogger is traveling at a constant rate of speed.

slope (speed) = rise (distance)run (time) = 1 mi

7 min

The jogger travels at a rate of 1 mile every 7minutes.

The amount of distance is the rise, and the amount of time is the run. You can find the speed by finding the slope.

5-6 Slope and Rates of Change

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

5-6 Slope and Rates of Change

Lesson Quiz: Part I

1. Tell whether the slope is positive or negative. Then find the slope.

Negative; -1

5-6 Slope and Rates of Change

Lesson Quiz: Part II

2. Use the slope and the point (–2, –3) to graph

the line.

12

5-6 Slope and Rates of Change

Lesson Quiz: Part III

3. Tell whether the graph shows a constant or

variable rate of change.

variable

5-6 Slope and Rates of Change

1. Tell whether the slope is positive or negative. Then identify the slope.

A. positive; 1

B. positive; 2

C. negative; –1

D. negative; –2

Lesson Quiz for Student Response Systems

5-6 Slope and Rates of Change

2. Use the slope and the point (–2, –3) to

identify the graph of the line.

A. B.

Lesson Quiz for Student Response Systems

1 4

5-6 Slope and Rates of Change

3. Which of the following graphs represents a variable rate of change?

A. B.

Lesson Quiz for Student Response Systems