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


Warm UpGraph the linear function.

y = 2x + 4


Problem of the Day

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

133 and 143


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


Vocabulary

slope

rate 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


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.



Additional Example 1A Continued

The rise is 3. The run is 3.

slope = riserun

= 33

= 1

3

3



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



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


Check It Out: Example 1A

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



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



Check It Out: Example 1B

(0, –4)

(–2, 4)

8

–2

The rise is 8. The run is –2.

slope = riserun

= 8–2 = –4


The line falls from left to right.

The slope is negative.


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


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

●

●


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!


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

●


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

●

●


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

●


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.


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.


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


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


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.


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


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.


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


Lesson Quiz: Part I

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

Negative; -1


Lesson Quiz: Part II

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

the line.

12


Lesson Quiz: Part III

3. Tell whether the graph shows a constant or

variable rate of change.

variable


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



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

identify the graph of the line.

A. B.


1 4


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

A. B.


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

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