9. A.FGR.2.2: Terry’s class is selling candy to raise money for a school trip. They’re in debt of $40 and selling each candy bar for $2. The profit represents the remaining amount of money after covering the cost of the candy through the candy bar sales. a. Graph the linear function representing the relationship between the number of candy bars and the profit. b. Identify the x-intercept and the y-intercept of the graph. c. Determine the domain and range of the function. d. Identify the interval where the function is increasing or decreasing. e. Describe the end behavior of the function. 

Let's tackle this problem step-by-step:

### a. Graph the Linear Function

First, we need to establish the linear function that represents the relationship between the number of candy bars sold ($x$) and the profit ($P$).

1. **Profit Calculation**: The profit is determined by the revenue from selling the candy bars minus the initial debt. Since each candy bar is sold for $2, the revenue from selling $x$ candy bars is $2x$. The debt is $40, so the profit function can be expressed as:
   \[
   P(x) = 2x - 40
   \]
   This is a linear equation where:
   - The slope is 2 (indicating the profit increases by $2 for each candy bar sold).
   - The y-intercept is $-40$ (indicating a debt of $40 when no candy bars are sold).

2. **Graphing the Function**: To graph the function, plot the y-intercept at $(0, -40)$ and use the slope to plot additional points.

### b. Identify the x-intercept and y-intercept of the Graph

- **Y-intercept**: The y-intercept occurs when $x = 0$. From the equation $P(x) = 2x - 40$, substituting $x = 0$:
  \[
  P(0) = 2(0) - 40 = -40
  \]
  So, the y-intercept is $(0, -40)$.

- **X-intercept**: The x-intercept occurs when $P(x) = 0$. Solving $2x - 40 = 0$:
  \[
  2x = 40 \implies x = 20
  \]
  So, the x-intercept is $(20, 0)$.

### c. Determine the Domain and Range of the Function

- **Domain**: The domain represents the possible values of $x$. Since you can't sell a negative number of candy bars, $x$ must be non-negative:
  \[
  \text{Domain: } x \geq 0
  \]

- **Range**: The range represents the possible values of the profit $P(x)$. Since the profit can be negative (indicating debt) or positive (indicating profit), the range starts from $-40$ and increases as more candy bars are sold:
  \[
  \text{Range: } P(x) \geq -40
  \]

### d. Identify the Interval Where the Function is Increasing or Decreasing

- The function is a straight line with a positive slope of 2, which means it is increasing across its entire domain:
  \[
  \text{The function is increasing for all } x \geq 0.
  \]

### e. Describe the End Behavior of the Function

- **As $x \to \infty$**: The profit $P(x) \to \infty$. This indicates that as the number of candy bars sold increases indefinitely, the profit will also increase indefinitely.

- **As $x \to 0$**: The profit approaches $-40$, indicating the initial debt when no candy bars are sold.

Would you like me to graph this function for you or do you have any specific part you'd like more details on?

Here are some related questions:

1. How would the graph change if the selling price per candy bar were increased to $3?
2. What if the initial debt was only $20? How would the x- and y-intercepts change?
3. How many candy bars would need to be sold to achieve a profit of $100?
4. If the cost per candy bar was $0.50 instead of $0, how would that affect the profit equation?
5. How would the domain and range change if the number of candy bars sold was capped at 50?

**Tip:** When identifying intercepts, always set the opposite variable to zero to find the intercepts accurately!

Terry’s class is selling candy to raise money for a school trip. They’re in debt of $40 and selling each candy bar for $2. The profit represents the remaining amount of money after covering the cost of the candy through the candy bar sales. a. Graph the linear function representing the relationship between the number of candy bars and the profit. b. Identify the x-intercept and the y-intercept of the graph. c. Determine the domain and range of the function. d. Identify the interval where the function is increasing or decreasing. e. Describe the end behavior of the function.

Explore how Terry’s class calculates profit by selling candy bars, including graphing a linear function, identifying intercepts, and understanding domain and range. Learn about the relationship between the number of candy bars sold and profit, suitable for students in Grades 7-9.

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