To determine the amount of money on the gift card when Nevaeh received it, we need to analyze the data provided. We have:
\[ \text{Coffee Cups (r)}: 3, 4, 6, 8 \]
\[ \text{Money Remaining (y)}: 21, 18, 12, 6 \]
From the data, we can see how much money is deducted for each cup of coffee:
- From 3 to 4 cups: 21 to 18, a drop of $3
- From 4 to 6 cups: 18 to 12, a drop of $6
- From 6 to 8 cups: 12 to 6, a drop of $6
It looks like each cup of coffee (after the first few) costs $3 based on the change observed. We are also noticing a linear relationship, which we can define with the formula:
\[ y = mx + b \]
Where:
- \( m \) is the slope (the change in money for each cup).
- \( b \) is the y-intercept (the initial amount of money on the gift card).
To find the equation of the linear function and the initial amount, we can use the slope, which appears constant at \(-3\) (since each cup costs 3 dollars). Knowing that the y-intercept represents the initial money amount when \( r = 0 \), we can use one of the points (let's use \( r = 3, y = 21 \)) to find \( b \):
- Substitute \( r \) and \( y \) into the equation: \[ 21 = -3(3) + b \] \[ 21 = -9 + b \] \[ b = 21 + 9 = 30 \]
So, the y-intercept is 30, meaning Nevaeh initially had $30 on the gift card.
The equation for the function is:
\[ y = -3r + 30 \]
To summarize:
- The amount of money on the gift card when Nevaeh received it: $30
- The equation to the function: \( y = -3r + 30 \)