1. Lesson 4.3 Graphing Proportional Relationships
11/13/2018
Lesson 4.3 Graphing Proportional Relationships
Students will be able to use a graph to identify if a relationship is proportional or not.

2. Proportions Example: 4 5 = 8 10
A proportion is an equation stating two ratios are equivalent (equal) or they have a constant rate.
Example: 4 5 = 8 10

3. Nonproportional Example: 4 5 ≠ 8 15
Relationships are nonproportional if the rates or ratios are not constant, equivalent.
Example: 4 5 ≠ 8 15

4. Unit Rate: A rate with a denominator of 1
Example:
4 𝑜𝑟𝑎𝑛𝑔𝑒𝑠 $1 ; 3 𝑏𝑜𝑦𝑠 1 𝑔𝑖𝑟𝑙

5. 24 𝑔𝑖𝑟𝑙𝑠 8 𝑏𝑜𝑦𝑠 $12.50 6 𝑜𝑟𝑎𝑛𝑔𝑒𝑠 14 𝑐𝑎𝑡𝑠 2 𝑙𝑏𝑠 𝑜𝑓 𝑓𝑜𝑜𝑑
Examples: Find the unit rate of each problem
24 𝑔𝑖𝑟𝑙𝑠 8 𝑏𝑜𝑦𝑠
$ 𝑜𝑟𝑎𝑛𝑔𝑒𝑠
14 𝑐𝑎𝑡𝑠 2 𝑙𝑏𝑠 𝑜𝑓 𝑓𝑜𝑜𝑑

6. Direct Variation: y = kx

7. 6 2 = 12 4 = 18 6
Simplify
3 1 = 3 1 = 3 1
10 2 = 16 4 = 22 6
Simplify
5 1 ≠ 4 1 ≠ 11 3

8. To be proportional: 1) must be a straight line and 2) the line must pass through the origin (0,0)

9. Since the simplified ratios were equal,
11/13/2018
Example 1: The following chart shows how much money Alex earns for mowing lawns. Is the amount of money he earns proportional to the number of hours that he spends mowing?
Earnings ($)
Hours (h)
Unit Rate ( ) 14 1 28 2 42 3 56 4
Since the simplified ratios were equal,
this was a proportional relationship.

10. Let’s graph this proportional relationship from Ex. 1 on an xy-plane.
We typically put time (hours) on the x-axis, and the earnings ($) on the y-axis.
Set up the graph paper to fit the data in the chart.
Plot points (x, y) from the table. y
Hours (h)
Earnings ($)
Point (x, y) 1 14
(1, 14) 2 28
(2, 28) 3 42
(3, 42) 4 56
(4, 56) 56 42
Earnings ($) 28 14
Connect the points. x 1 2 3 4 5
Describe the graph of this
proportional relationship.
Hours worked

11. between cost and the number of tickets ordered.
Example 2: Ticket Express charges $7 per movie ticket plus a $3 processing fee per order. Is the cost of an order proportional to the number of tickets ordered? Explain .
Cost ($) 10 17 24 31
Tickets Ordered 1 2 3 4
Since all of the simplified ratios are not equal, there is NOT a proportional relationship
between cost and the number of tickets ordered.

12. Now, let’s graph this nonproportional relationship from Ex. 2.
Tickets ordered will be on the x-axis,
and the cost ($) will be on the y-axis. y
Plot points (x, y) from the table. 32
Tickets
Earnings ($)
Point (x, y) 1 10
(1, 10) 2 17
(2, 17) 3 24
(3, 24) 4 31
(4, 31) 28 24
Cost ($) 20 16 12 8 4
Connect the points. x
Describe the graph of this
nonproportional relationship. 1 2 3 4
Tickets ordered

13. Ex. 1 Write an equation in function notation for the graph.

14. Ex. 1 Write an equation in function notation for the graph.y

15. Ex. 1 Write an equation in function notation for the graph.y -1 -2

16. Ex. 1 Write an equation in function notation for the graph.y -1 -2

17. Ex. 1 Write an equation in function notation for the graph.y -1 -2 1 2

18. Ex. 1 Write an equation in function notation for the graph.y -1 -2 1 2 4

19. Ex. 1 Write an equation in function notation for the graph.y -1 -2 1 2 4

20. Ex. 1 Write an equation in function notation for the graph.
Goes through origin so
direct variation & proportional x y -1 -2 1 2 4

21. Ex. 1 Write an equation in function notation for the graph.
Goes through origin so
direct variation & proportional x y -1 -2 1 2 4 +1 +1 +1

22. Ex. 1 Write an equation in function notation for the graph.
Goes through origin so
direct variation & proportional x y -1 -2 1 2 4 +1 +2 +1 +2 +1 +2

23. Ex. 1 Write an equation in function notation for the graph.
Goes through origin so
direct variation & proportional
y = 2x x y -1 -2 1 2 4 +1 +2 +1 +2 +1 +2

24. Ex. 1 Write an equation in function notation for the graph.
Goes through origin so
direct variation & proportional
y = 2x
f(x) = 2x x y -1 -2 1 2 4 +1 +2 +1 +2 +1 +2

25. Ex. 2 Write an equation in function notation for the graph.

26. Ex. 2 Write an equation in function notation for the graph.y -1 -3 1 2 3

27. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional x y -1 -3 1 2 3

28. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional x y -1 -3 1 2 3 +1 +2 +1 +2 +1 +2

29. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional x 2x y -1 -3 1 2 3 +1 +2 +1 +2 +1 +2

30. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional x 2x y -1 -2 -3 1 2 4 3 +1 +2 +1 +2 +1 +2

31. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional
y = 2x x 2x y -1 -2 -3 1 2 4 3 +1 +2 +1 +2 +1 +2

32. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional
y = 2x x 2x y -1 -2 -3 1 2 4 3 +1 +2 +1 +2 +1 +2

33. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional
y = 2x x 2x y -1 -2 -3 1 2 4 3 -1 +1 +2 -1 +1 +2 -1 +1 +2 -1

34. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional
y = 2x – 1 x 2x y -1 -2 -3 1 2 4 3 -1 +1 +2 -1 +1 +2 -1 +1 +2 -1

35. Ex. 2 Write an equation in function notation for the graph.
DOES NOT GO through origin
so Nonproportional
y = 2x – 1
f(x) = 2x – 1 x 2x y -1 -2 -3 1 2 4 3 -1 +1 +2 -1 +1 +2 -1 +1 +2 -1

36. Ex. 3 The total snowfall each hour of a winter
Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below.
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60

37. Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below. a. Write an equation for the data.
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60

38. Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below. a. Write an equation for the data. +1 +1 +1
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60

39. Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below. a. Write an equation for the data. +1 +1 +1
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60
+1.65
+1.65
+1.65

40. Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below. a. Write an equation for the data. y = 1.65x +1 +1 +1
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60
+1.65
+1.65
+1.65

41. Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below. a. Write an equation for the data. y = 1.65x b. Describe the relationship between the hour and inches of snowfall. +1 +1 +1
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60
+1.65
+1.65
+1.65

42. Ex. 3 The total snowfall each hour of a winter snowstorm is shown in the table below. a. Write an equation for the data. y = 1.65x b. Describe the relationship between the hour and inches of snowfall. Proportional +1 +1 +1
Hour 1 2 3 4
Inches of Snowfall
1.65
3.30
4.95
6.60
+1.65
+1.65
+1.65