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.

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

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

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

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 $12.50 6 𝑜𝑟𝑎𝑛𝑔𝑒𝑠 14 𝑐𝑎𝑡𝑠 2 𝑙𝑏𝑠 𝑜𝑓 𝑓𝑜𝑜𝑑

Direct Variation: y = kx

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

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

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.

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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