1. Find the x and y intercepts: y = 7x – 3 -4y + x = 8 2x – 5y = 20
Lesson 3.4 Find Slope and Rate of Change Essential Question: How do you find slope of a line and interpret slope as a rate of change?
Warm-up:
Find the x and y intercepts:
y = 7x – 3
-4y + x = 8
2x – 5y = 20
Graph: x = -2
Graph: y = 3
Common
Core
CC.9-12.F.IF.6
Graph linear and quadratic functions and show intercepts, maxima, and minima.

9. Finding the slope of a line

11. Find the slope of the line shown.
EXAMPLE 1
Find a positive slope
Find the slope of the line shown.
Let (x1, y1) = (–4, 2) = (x2, y2) = (2, 6).
m =
y2 – y1
x2 – x1
Write formula for slope.
6 – 2
2 – (–4) =
Substitute. = 4 6 2 3
Simplify.

12. EXAMPLE 2
GUIDED PRACTICE
Write an equation from a graph
for Example 1
Find the slope of the line that passes through the points.
1. (5, 2) and (4, –1)
ANSWER 3

13. Write an equation from a graph
GUIDED PRACTICE
for Example 1
Find the slope of the line that passes through the points.
2. (–2, 3) and (4, 6) 1 2
ANSWER

14. Write an equation from a graph
GUIDED PRACTICE
for Example 1
Find the slope of the line that passes through the points.
3. ( , 5) and ( , –3) 9 2 1
ANSWER 2

15. Find the slope of the line shown.
EXAMPLE 2
Find a negative slope
Find the slope of the line shown.
Let (x1, y1) = (3, 5) and (x2, y2) = (6, –1).
m =
y2 – y1
x2 – x1
Write formula for slope.
–1 – 5
6 – 3 =
Substitute.
– 6 3 = –2
Simplify.

16. Find the slope of a horizontal line
EXAMPLE 3
Find the slope of a horizontal line
Find the slope of the line shown.
Let (x1, y1) = (–2, 4) and (x2, y2) = (4, 4).
m =
y2 – y1
x2 – x1
Write formula for slope.
4 – 4
4 – (–2) =
Substitute. 6 =
Simplify.

17. Find the slope of a vertical line
EXAMPLE 4
Find the slope of a vertical line
Find the slope of the line shown.
Let (x1, y1) = (3, 5) and (x2, y2) = (3, 1).
m =
y2 – y1
x2 – x1
Write formula for slope.
1 – 5
3 – 3 =
Substitute.
– 4 =
Division by zero is undefined.
ANSWER
Because division by zero is undefined, the slope of a vertical line is undefined.

18. EXAMPLE 2
Write an equation from a graph
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the slope of the line that passes through the points.
4. (5, 2) and (5, –2)
ANSWER
undefined

19. Write an equation from a graph
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the slope of the line that passes through the points.
5. (0, 4) and (–3, 4)
ANSWER

20. Write an equation from a graph
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the slope of the line that passes through the points.
6. (0, 6) and (5, –4)
ANSWER –2

21. EXAMPLE 5
Find a rate of change
INTERNET CAFE
The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time.
Time (hours) 2 4 6
Cost (dollars) 7 14 21

22. EXAMPLE 5
Find a rate of change
SOLUTION =
change in cost
change in time
Rate of change
14 – 7
4 – 2 = 7 2
3.5
ANSWER
The rate of change in cost is $3.50 per hour.

23. GUIDED PRACTICE
for Example 5
The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. 7.
EXERCISE
Time(minute) 30 60 90
Distance (miles)
1.5 3
4.5
ANSWER
0.05 mi/min

24. EXAMPLE 6
Use a graph to find and compare rates of change
COMMUNITY THEATER
A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time.

25. Use a graph to find and compare rates of change
EXAMPLE 6
Use a graph to find and compare rates of change
SOLUTION
Find the rates of change using the slope formula.
232 – 124
4 – 1 =
108 3
Weeks 1–4:
= 36 people per week
204 – 232
6 – 4 =
–28 2
Weeks 4–6:
= –14 people per week
72 – 204
10 – 6 =
–132 4
Weeks 6–10:
= –33 people per week
ANSWER
Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly.

26. EXAMPLE 7
Interpret a graph
COMMUTING TO SCHOOL
A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words.

27. EXAMPLE 7
Interpret a graph
SOLUTION
The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus.

28. EXAMPLE 7
GUIDED PRACTICE
Interpret a graph
for Examples 6 and 7
WHAT IF? How would the answer to Example 6 change if you knew that attendance was 70 people in week 12? 8.
Sample answer: The attendance did not decrease as rapidly between weeks 10 and 12.
ANSWER

29. EXAMPLE 7
GUIDED PRACTICE
Interpret a graph
for Examples 6 and 7
WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. 9.
ANSWER

30. Classwork/Homework
4.4 Practice B