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How would I solve this with a graphing calculator 

To solve this problem with a graphing calculator, here are the steps you can follow. The given function is:

\[
f(x) = -0.005x^2 + 120x - 100,000
\]

This function represents the profit based on the number of items \( x \) produced and sold. You are asked to find the number of items \( x \) that maximizes profit, which corresponds to finding the vertex of this quadratic function.

### Steps on a Graphing Calculator (TI-84 as an example):

1. **Input the Function**:
   - Turn on the graphing calculator.
   - Press the `Y=` button.
   - Enter the equation: \(-0.005x^2 + 120x - 100000\) in the form of `Y1=`.

2. **Set the Window**:
   - Press the `WINDOW` button to adjust the viewing window. Based on the context (the number of items should be positive), you can set:
     - `Xmin = 0` (since the company can't sell negative items).
     - `Xmax = 30,000` (to ensure the graph is visible within a reasonable range of items).
     - `Ymin = -200,000` (to give space for possible negative profits).
     - `Ymax = 200,000` (to allow a view of the positive profit area).
   
3. **Graph the Function**:
   - Press the `GRAPH` button to display the function.
   
4. **Find the Maximum**:
   - Since this is a quadratic equation with a downward-facing parabola (because of the negative coefficient of \(x^2\)), the maximum profit will occur at the vertex.
   - Press the `2nd` button, then `TRACE` to access the `CALC` menu.
   - Select option `4: Maximum`.
   - Move the cursor using the left and right arrow keys to the left of the peak of the graph and press `ENTER`.
   - Move the cursor to the right of the peak and press `ENTER` again.
   - The calculator will ask for a guess—move the cursor close to the peak and press `ENTER`.
   - The calculator will display the \(x\) value of the maximum, which is the number of items that maximizes the profit, and the \(y\) value, which represents the maximum profit.

5. **Interpret the Result**:
   - The \(x\)-value given is the number of items the company should produce and sell to maximize their profit.
   - Based on the function provided, this value should fall within one of the multiple-choice intervals (A-E) on the problem.

Would you like further assistance with interpreting the result? 

### Relative questions:
1. What is the significance of the vertex in a quadratic function related to profit maximization?
2. How can you determine whether a quadratic function has a maximum or minimum?
3. Why is it important to set a proper window for graphing in this scenario?
4. What does the negative coefficient of \(x^2\) indicate about the profit function's graph?
5. How can you use the graph to estimate the profit for a different number of items?

**Tip**: Always check your graphing window to ensure that both \(x\) and \(y\) values make sense in the context of the problem, especially for real-world applications like profit analysis.

How would I solve this with a graphing calculator?

Learn how to solve a quadratic function to maximize profit using a graphing calculator. This problem involves understanding key algebraic concepts such as quadratic functions and vertex calculation. Suitable for high school students.

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