https://file.aiquickdraw.com/mathbot/file/172990727399861tfkqk9.jpg

Let's analyze the content of the image and solve each part step by step.

### Information from the Table
The table shows:
- **Number of Computers Made:** 5, 15, 25, 35, 45, 55, 65
- **Profit for the Company ($):** 0, 5000, 8000, 9000, 8000, 5000, 0

### Questions:
#### (a) Plot the points and connect them to graph the underlying function. 
We need to determine which graph (A, B, C, or D) represents this data.

Let's analyze the given data points:
- (5, 0)
- (15, 5000)
- (25, 8000)
- (35, 9000)
- (45, 8000)
- (55, 5000)
- (65, 0)

Observing these points, the profit increases as the number of computers increases, reaches a peak at 35 computers, and then decreases symmetrically. This suggests a **parabolic shape**. Looking at the graphs provided, we need to find one that matches this pattern.

**Correct Answer for Part (a):** The graph is **C**. This graph represents a parabolic function that peaks at 35 computers.

#### (b) How many computers should be made each day to achieve the maximum profit?
From the table, the maximum profit of $9000 is achieved when the company makes **35 computers**.

**Correct Answer for Part (b):** **35 computers**

#### (c) If the company wants to earn a profit of $8000 or more each day, how many computers should they build each day?
From the data, a profit of $8000 is achieved when:
- 25 computers are made (increasing side)
- 45 computers are made (decreasing side)

Thus, the company should build between **25 and 45 computers** to ensure a profit of $8000 or more.

**Correct Answer for Part (c):** Between **25 and 45 computers**

#### (d) What will the profit picture be if the company builds 67 computers per day?
Looking at the data, the trend shows that the profit returns to 0 when the company makes 65 computers. Building **67 computers** would likely mean the company continues to experience a loss.

**Correct Answer for Part (d):** **The company will have a loss.**

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Do you want any more detailed explanations or have any questions? Here are some related questions to think about:

1. How would the graph change if the maximum profit shifted to 45 computers?
2. What type of mathematical function typically represents this kind of profit data?
3. How can we determine the symmetry of a parabola from data points?
4. What if the profit values were slightly different; would that affect the maximum point?
5. Can you determine the equation of the parabola using the data points?

**Tip:** When looking at graphs, pay attention to the shape and symmetry. Symmetry often gives clues about the type of function being represented (e.g., parabolas are symmetric around their vertex).

The table above shows the relationship between the number of computers built by a company each day and the company's profit for that day. (a) Plot the points and connect them to graph the underlying function. Which of the graphs in the window on the right is the graph of the data shown? (b) How many computers should be made each day to achieve the maximum profit? (c) If the company wants to earn a profit of $8000 or more each day, how many computers should they build each day? (d) What will the profit picture be if the company builds 67 computers per day?

Explore how to determine maximum profit and optimal production level based on quadratic functions. This problem involves analyzing data points to plot a parabolic function, finding maximum values, and interpreting profit trends based on production quantities.

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