1. Rate of Change and Slope
4-3
Rate of Change and Slope
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
Holt McDougal Algebra 1

2. Warm Up 1. Find the x- and y-intercepts of 2x – 5y = 20.
Describe the correlation shown by the scatter plot. 2.
x-int.: 10; y-int.: –4
negative

3. Objectives Find rates of change and slopes.
Relate a constant rate of change to the slope of a line.

4. Vocabulary
rate of change
rise
run
slope

5. 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.

6. Step 1 Identify the dependent and independent variables.
Example 1: Application
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?
Step 1 Identify the dependent and independent variables.
dependent: temperature independent: month

7. Example 1 Continued
Step 2 Find the rates of change.
2 to 3
3 to 5
5 to 7
7 to 8
The temperature increased at the greatest rate from month 5 to month 7.

8. Step 1 Identify the dependent and independent variables.
Check It Out! Example 1
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?
Step 1 Identify the dependent and independent variables.
dependent: balance independent: day

9. Check It Out! Example 1 Continued
Step 2 Find the rates of change.
1 to 6
6 to 16
16 to 22
22 to 30
The balance declined at the greatest rate from day 1 to day 6.

10. Example 2: Finding Rates of Change from a Graph
Graph the data from Example 1 and show the rates of change.
Graph the ordered pairs. The vertical segments show the changes in the dependent variable, and the horizontal segments show the changes in the independent variable.
Notice that the greatest rate of change is represented by the steepest of the red line segments.

11. Example 2 Continued
Graph the data from Example 1 and show the rates of change.
Also notice that between months 2 to 3, when the balance did not change, the line segment is horizontal.

12. Check It Out! Example 2
Graph the data from Check It Out Example 1 and show the rates of change.
Graph the ordered pairs. The vertical segments show the changes in the dependent variable, and the horizontal segments show the changes in the independent variable.
Notice that the greatest rate of change is represented by the steepest of the red line segments.

13. Check It Out! Example 2 Continued
Graph the data from Check It Out Problem 1 and show the rates of change.
Also notice that between days 16 to 22, when the balance did not change, the line segment is horizontal.

14. If all of the connected segments have the same rate of change, then they all have the same steepness and together form a straight line. The constant rate of change of a line is called the slope of the line.

16. Example 3: Finding Slope
Find the slope of the line.
Run –9
Begin at one point and count vertically to fine the rise.
(–6, 5) • •
Rise –3
Run 9
Then count horizontally to the second point to find the run.
Rise 3
(3, 2)
It does not matter which point you start with. The slope is the same.

17. Find the slope of the line that contains (0, –3) and (5, –5).
Check It Out! Example 3
Find the slope of the line that contains (0, –3) and (5, –5).
Begin at one point and count vertically to find rise.
Then count horizontally to the second point to find the run.
Run –5
It does not matter which point you start with. The slope is the same. •
Rise –2
Rise 2 •
Run 5

18. Example 4: Finding Slopes of Horizontal and Vertical Lines
Find the slope of each line. A. B.
You cannot divide by 0
The slope is undefined.
The slope is 0.

19. Check It Out! Example 4
Find the slope of each line.
4a.
4b.
You cannot divide by 0.
The slope is undefined.
The slope is 0.

20. As shown in the previous examples, slope can be positive, negative, zero or undefined. You can tell which of these is the case by looking at a graph of a line–you do not need to calculate the slope.

21. Example 5: Describing Slope
Tell whether the slope of each line is positive, negative, zero or undefined. A. B.
The line rises from left to right.
The line falls from left to right.
The slope is positive.
The slope is negative.

22. Check It Out! Example 5
Tell whether the slope of each line is positive, negative, zero or undefined. a. b.
The line is vertical.
The line rises from left to right.
The slope is positive.
The slope is undefined.

24. Lesson Quiz: Part I
Name each of the following.
1. The table shows the number of bikes made by a company for certain years. Find the rate of change for each time period. During which time period did the number of bikes increase at the fastest rate?
1 to 2: 3; 2 to 5: 4; 5 to 7: 0; 7 to 11: 3.5;
from years 2 to 5

25. Lesson Quiz: Part II
Find the slope of each line. 2. 3.
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26. 4-4 The Slope Formula Warm Up Lesson Presentation Lesson Quiz
Holt Algebra 1
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1

27. Warm Up Add or subtract. 1. 4 + (–6) 2. –3 + 5 3. –7 – 7 4. 2 – (–1)
(–6) –3 + 5
3. –7 – – (–1) –2 2
–14 3
Find the x- and y-intercepts.
5. x + 2y = 8
6. 3x + 5y = –15
x-intercept: 8; y-intercept: 4
x-intercept: –5; y-intercept: –3

28. Objective
Find slope by using the slope formula.

29. In Lesson 5-3, slope was described as the constant rate of change of a line. You saw how to find the slope of a line by using its graph.
There is also a formula you can use to find the slope of a line, which is usually represented by the letter m. To use this formula, you need the coordinates of two different points on the line.

31. Example 1: Finding Slope by Using the Slope Formula
Find the slope of the line that contains (2, 5) and (8, 1).
Use the slope formula.
Substitute (2, 5) for (x1, y1) and (8, 1) for (x2, y2).
Simplify.
The slope of the line that contains (2, 5) and (8, 1) is

32. Check It Out! Example 1a
Find the slope of the line that contains (–2, –2) and (7, –2).
Use the slope formula.
Substitute (–2, –2) for (x1, y1) and (7, –2) for (x2, y2).
Simplify.
= 0
The slope of the line that contains (–2, –2) and (7, –2) is 0.

33. Check It Out! Example 1b
Find the slope of the line that contains (5, –7) and (6, –4).
Use the slope formula.
Substitute (5, –7) for (x1, y1) and (6, –4) for (x2, y2).
Simplify.
= 3
The slope of the line that contains (5, –7) and (6, –4) is 3.

34. Check It Out! Example 1c
Find the slope of the line that contains and
Use the slope formula.
Substitute for (x1, y1) and for (x2, y2) and simplify.
The slope of the line that contains and is 2.

35. Sometimes you are not given two points to use in the formula
Sometimes you are not given two points to use in the formula. You might have to choose two points from a graph or a table.

36. Example 2A: Finding Slope from Graphs and Tables
The graph shows a linear relationship. Find the slope.
Let (0, 2) be (x1, y1) and (–2, –2) be (x2, y2).
Use the slope formula.
Substitute (0, 2) for (x1, y1) and (–2, –2) for (x2, y2).
Simplify.

37. Example 2B: Finding Slope from Graphs and Tables
The table shows a linear relationship. Find the slope.
Step 1 Choose any two points from the table. Let (0, 1) be (x1, y1) and (–2, 5) be (x2, y2).
Step 2 Use the slope formula.
Use the slope formula.
Substitute (0, 1) for
and (–2, 5) for .
Simplify.
The slope equals –2

38. The graph shows a linear relationship. Find the slope.
Check It Out! Example 2a
The graph shows a linear relationship. Find the slope.
Let (4, 4) be (x1, y1) and (8, 6) be (x2, y2).
Use the slope formula.
Substitute (4, 4) for (x1, y1) and (8, 6) for (x2, y2).
Simplify.

39. The graph shows a linear relationship. Find the slope.
Check It Out! Example 2b
The graph shows a linear relationship. Find the slope.
Let (–2, 4) be (x1, y1) and (0, –2) be (x2, y2).
Use the slope formula.
Substitute (–2, 4) for (x1, y1) and (0, –2) for (x2, y2).
Simplify.

40. The table shows a linear relationship. Find the slope.
Check It Out! Example 2c
The table shows a linear relationship. Find the slope.
Step 1 Choose any two points from the table. Let (0, 1) be (x1, y1) and (2, 5) be (x2, y2).
Step 2 Use the slope formula.
Use the slope formula.
Substitute (0, 1) for (x1, y1) and (2, 5) for (x2, y2).
Simplify.

41. The table shows a linear relationship. Find the slope.
Check It Out! Example 2d
The table shows a linear relationship. Find the slope.
Step 1 Choose any two points from the table. Let (0, 0) be (x1, y1) and (–2, 3) be (x2, y2).
Step 2 Use the slope formula.
Use the slope formula.
Substitute (0, 0) for (x1, y1) and (–2, 3) for (x2, y2).
Simplify

42. Remember that slope is a rate of change
Remember that slope is a rate of change. In real-world problems, finding the slope can give you information about how a quantity is changing.

43. Example 3: Application
The graph shows the average electricity costs (in dollars) for operating a refrigerator for several months. Find the slope of the line. Then tell what the slope represents.
Step 1 Use the slope formula.

44. Example 3 Continued
Step 2 Tell what the slope represents.
In this situation y represents the cost of electricity and x represents time.
So slope represents in units of .
A slope of 6 mean the cost of running the refrigerator is a rate of 6 dollars per month.

45. Check It Out! Example 3
The graph shows the height of a plant over a period of days. Find the slope of the line. Then tell what the slope represents.
Step 1 Use the slope formula.

46. Check It Out! Example 3
Step 2 Tell what the slope represents.
In this situation y represents the height of the plant and x represents time.
So slope represents in units of .
A slope of mean the plant grows at rate of 1 centimeter every two days.

47. If you know the equation that describes a line, you can find its slope by using any two ordered-pair solutions. It is often easiest to use the ordered pairs that contain the intercepts.

48. Example 4: Finding Slope from an Equation
Find the slope of the line described by 4x – 2y = 16.
Step 1 Find the x-intercept.
Step 2 Find the y-intercept.
4x – 2y = 16
4x – 2y = 16
4x – 2(0) = 16
Let y = 0.
4(0) – 2y = 16
Let x = 0.
4x = 16
x = 4
–2y = 16
y = –8
Step 3 The line contains (4, 0) and (0, –8). Use the slope formula.

49. Check It Out! Example 4
Find the slope of the line described by 2x + 3y = 12.
Step 1 Find the x-intercept.
Step 2 Find the y-intercept.
2x + 3y = 12
2x + 3y = 12
2x + 3(0) = 12
Let y = 0.
2(0) + 3y = 12
Let x = 0.
2x = 12
x = 6
3y = 12
y = 4
Step 3 The line contains (6, 0) and (0, 4). Use the slope formula.

50. Lesson Quiz
1. Find the slope of the line that contains (5, 3) and (–1, 4).
2. Find the slope of the line. Then tell what the slope represents.
50; speed of bus is 50 mi/h
3. Find the slope of the line described by x + 2y = 8.