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A graph shows a decrease in amount of money on a gift card and how many coffee's were spent using the card as it starts on 25 and slowly decreases until 10 coffee's were bought and left the money at 0

a) Is this a proportional relationship?
Yes, because it's a straight line

b) What is the y-intercept?
(0, 25)

c) What does the y-intercept mean in context of the scenario?
The starting value of the gift card before buying any coffee is $25

d) What is the slope?

e) What does the slope mean in context of the scenario?
The starting value of the gift card before buying any coffee is $25

3 answers

Let's analyze the scenario step by step and answer parts d and e regarding the slope.

a) Is this a proportional relationship?

  • You mentioned it is a straight line, which is a good indicator. However, for it to be proportional, it should pass through the origin (0,0), meaning if no coffees are bought (0), there should still be $0 remaining on the gift card. Since the y-intercept is $25, it is not a proportional relationship.

b) What is the y-intercept?

  • Correct, the y-intercept is (0, 25). This point indicates the starting money on the gift card.

c) What does the y-intercept mean in context of the scenario?

  • Correct, it means the starting value of the gift card before buying any coffee is $25.

d) What is the slope?

  • To find the slope, you would take the change in money divided by the change in the number of coffees.
    • Assuming each coffee costs the same amount, let's denote the cost per coffee as \( m \).
    • When 10 coffees are bought, the card value decreases from $25 to $0.
    • Thus, if the total cost for 10 coffees equals $25: \[ \text{Cost per coffee} = \frac{25}{10} = 2.5 \]
    • Thus, the slope \( m \) is -2.5, indicating that for every coffee bought, the card's value decreases by $2.50.

e) What does the slope mean in context of the scenario?

  • The slope of -2.5 indicates that for every coffee purchased, $2.50 is deducted from the gift card's balance. This shows how the money on the card decreases as coffees are purchased, reflecting the cost of each coffee.
The slope options: -5, +5/2, -5/2, 10, 25

Given the information you provided about the gift card and the number of coffees purchased, let’s re-evaluate the scenario with the context of the slope options provided.

  1. Starting amount on the gift card is $25.
  2. Amount left after 10 coffees is $0.

The total cost for 10 coffees is $25, meaning the cost per coffee can be calculated as follows:

\[ \text{Cost per coffee} = \frac{\text{Total cost}}{\text{Number of coffees}} = \frac{25}{10} = 2.5 \]

To find the slope (rate of decrease of money per coffee bought):

  • Each coffee costs $2.50, so the slope, which represents the rate of change of the gift card’s balance with respect to the number of coffees purchased, should be negative (as the money is decreasing):

\[ \text{Slope} = -\text{cost per coffee} = -2.5 \]

Looking at the options you provided:

  • -5
  • +5/2 (which is equal to +2.5)
  • -5/2 (which is equal to -2.5)
  • 10
  • 25

Since $-2.5$ is equivalent to \(-\frac{5}{2}\), the correct answer for the slope in context of your graph is:

d) The slope is \(-\frac{5}{2}\).

e) What does the slope mean in context of the scenario?

  • The slope \(-\frac{5}{2}\) indicates that for every coffee purchased, $2.50 is deducted from the balance of the gift card.