Answer:
Frank's plan: It's proportional.
Allie's plan: It's not proportional.
Step-by-step explanation:
When the independent and dependent variables are proportional, that is, when the independent variable increases, the dependent variable does so in the same proportion, and when the independent variable decreases, the dependent variable also does so in the same proportion, then the function that relates them is of direct proportionality.
Proportionality can be demonstrated by any of the following three conditions:
Condition 1: the slope of the line is equal to the coefficient of the independent variable.
Be,
y = dependent variable = Cost
x = independent variable = Time
m = slope = (y2 - y1) / (x2 - x1)
c = coefficient of the independent variable = x1 / x2
Frank's plan: It's proportional
m = (3.00 - 1.50) / (6 - 3) = 1.50 / 3 = 1/2
c = 3/6 = 1/2
m = c
The condition is met. It is proportional.
Allie's plan:
m = (5.00 - 4.50) / (6 - 3) = 0.50 / 3 = 1/6
c = 3/6 = 1/2
m ≠ c
The condition is not met. It is not proportional
Condition 2: the function is a line that passes through the origin point (0,0).
Frank's plan: it is proportional to the number of minutes, since the straight line of the function passes through the origin point (0,0).
Allie's plan: it is not proportional to the number of minutes because the line of the function does not pass through the point (0,0), but instead passes through the point (0,4).
Condition 3: The function does not carry an independent term.
Frank's plan:
y = mx + b, where m = 1/2 and b = 0
y = 1/2x
x = 0 ---> y = (1/2) * 0 = 0
x = 3 ---> y = (1/2) * 3 = 1.50
x = 6 ---> y = (1/2) * 6 = 3.00
There is no independent term in the function. It is proportional.
Allie's plan:
y = mx + b, where m = 1/6 and b = 4
y = 1/6x + 4
x = 0 ---> y = (1/6) * 0 + 4 = 0 + 4 = 4.00
x = 3 ---> y = (1/6) * 3 + 4 = 0.5 + 4 = 4.50
x = 6 ---> y = (1/6) * 6 + 4 = 1 + 4 = 5.00
The function has the independent term b = 4 so that the function passes through that point (0.4). Hence, it’s not proportional.
See attached file with the plotted functions.
Hope this helps!
