Question

nevaeh uses a gift card to buy a cup of coffee each morning. the table shows the amount of money left on the card, y, as a function of the number of cups of coffee she buys, x. \\(\boldsymbol{\text{find and describe the rate of change of the function.}}\\) the rate of change is \\(-3\\). it represents a decrease in the amount on the gift card for every cup nevaeh purchases. \\(\boldsymbol{\text{find and describe the initial value of the function.}}\\) the initial value is \\(28\\). \\(\

$$\begin{array}{|c|c|}\\hline \\text{number of cups of coffee, } x & \\text{amount of money left on card (\\$), } y \\\\ \\hline 2 & 19 \\\\ \\hline 5 & 10 \\\\ \\hline 7 & 4 \\\\ \\hline \\end{array}$$

\\)

Explanation:

Part (a): Rate of Change

Step1: Recall Rate of Change Formula

The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Choose Two Points

Take \((2, 19)\) and \((5, 10)\). Here, \(x_1 = 2\), \(y_1 = 19\), \(x_2 = 5\), \(y_2 = 10\).

Step3: Calculate Rate of Change

\(\frac{10 - 19}{5 - 2}=\frac{-9}{3}=-3\). This negative value means the amount on the card decreases by £3 for each cup of coffee bought.

Step1: Use Linear Equation Form

The linear function is \(y = mx + b\), where \(m=-3\) (rate of change), \(x\) is cups, \(y\) is balance, and \(b\) is initial value (when \(x = 0\)).

Step2: Substitute a Point

Use \((x, y)=(2, 19)\) and \(m=-3\) into \(y = mx + b\): \(19=-3(2)+b\).

Step3: Solve for \(b\)

\(19=-6 + b\Rightarrow b=19 + 6=25\)? Wait, the given answer is 28. Wait, let's check with another point. Use \((5, 10)\): \(10=-3(5)+b\Rightarrow10=-15 + b\Rightarrow b = 25\). Wait, maybe the table has a typo or miscalculation. But according to the provided answer, let's re - evaluate. Wait, maybe the rate of change was miscalculated. Wait, if we take \((2,19)\) and \((7,4)\): \(\frac{4 - 19}{7 - 2}=\frac{-15}{5}=-3\). Then using \((2,19)\): \(y=-3x + b\Rightarrow19=-6 + b\Rightarrow b = 25\). But the given initial value is 28. There must be an error in the problem or the provided answer. However, following the given answer's logic (maybe a different rate of change or point), if we assume the initial value is 28, it would mean when \(x = 0\) (no coffee bought), the card has £28. This represents the original amount on the gift card before any coffee was purchased.

Answer:

The rate of change is \(-3\), representing a decrease of £3 in the gift card balance per cup of coffee purchased.

Part (b): Initial Value