Question

homework practice
graph proportional relationships
for exercises 1 and 2, determine whether the relationship between the two quantities shown in each table are proportional by graphing on the coordinate plane. explain your reasoning.
1.

temperature (degrees)
0	32
5	41
10	50
15	59
20	68

2.

popcorn
0	0
1	4
2	8
3	12
4	16

1. movies an online dvd rental company charges $15 a month for unlimited rentals. determine whether the total paid after each month is proportional to number of months by graphing on the coordinate plane. explain your reasoning.

Explanation:

Exercise 1: Temperature Relationship

Step1: Identify Data Points

The table gives pairs: (0, 32), (5, 41), (10, 50), (15, 59), (20, 68).

Step2: Check for Proportionality

A proportional relationship has the form \( y = kx \) (passes through the origin, \( k = \frac{y}{x} \) constant). For \( x = 0 \), \( y = 32
eq 0 \), so the line does not pass through (0,0). Also, calculate \( \frac{y}{x} \):

• \( \frac{32}{0} \) (undefined), \( \frac{41}{5} = 8.2 \), \( \frac{50}{10} = 5 \) (not constant).

Step3: Graph Analysis

Plotting the points, the line does not pass through the origin, and the ratio \( \frac{\text{Fahrenheit}}{\text{Celsius}} \) is not constant. Thus, the relationship is not proportional.

Exercise 2: Popcorn Cost Relationship

Step1: Identify Data Points

The table gives pairs: (0, 0), (1, 4), (2, 8), (3, 12), (4, 16).

Step2: Check for Proportionality

A proportional relationship has \( y = kx \) (passes through origin, \( k \) constant). Here, \( \frac{\text{Cost}}{\text{Bags}} = \frac{4}{1} = \frac{8}{2} = \frac{12}{3} = \frac{16}{4} = 4 \) (constant \( k = 4 \)).

Step3: Graph Analysis

Plotting the points, the line passes through (0,0) and has a constant slope (ratio), so it is proportional.

Exercise 3: DVD Rental Cost Relationship

Step1: Define Variables

Let \( x \) = months, \( y \) = total cost (\$). The cost is LXI2 (since \$15 per month).

Step2: Check Proportionality

For \( x = 0 \), \( y = 0 \) (passes through origin). The ratio \( \frac{y}{x} = 15 \) (constant, since \( y = 15x \)).

Step3: Graph Analysis

Plotting points (0,0), (1,15), (2,30), etc., the line passes through the origin and has a constant slope (ratio), so it is proportional.

Answer:

s:

1. The relationship between Celsius and Fahrenheit is not proportional (does not pass through origin, ratio not constant).
2. The relationship between bags of popcorn and cost is proportional (passes through origin, constant ratio \( k = 4 \)).
3. The relationship between months and total DVD rental cost is proportional (passes through origin, constant ratio \( k = 15 \)).