Perform quadratic regression using a graphing calculator to determine a quadratic function that could be used to model the data in the table below.

A 7 by 2 data table.
Data table reads: StartLayout 1st Row 1st Column Bold Horizontal Distance in feet 2nd Column Bold Height in feet 2nd Row 1st Column 0 2nd Column 5 3rd Row 1st Column 10 2nd Column 8 4th Row 1st Column 20 2nd Column 12 5th Row 1st Column 30 2nd Column 20 6th Row 1st Column 40 2nd Column 15 7th Row 1st Column 50 2nd Column 10 EndLayout.

Round numbers in answer to 4 decimal places.

To perform quadratic regression using the given data and obtain a quadratic function that models the relationship between horizontal distance (in feet) and height (in feet), follow these steps:

### 1. Organize the dataTo perform quadratic regression using the given data and obtain a quadratic function that models the relationship between horizontal distance (in feet) and height (in feet), follow these steps:

### 1. Organize the 

| **Horizontal Distance (ft)** | **Height (ft)** |
|------------------------------|-----------------|
| 0                            | 5               |
| 10                           | 8               |
| 20                           | 12              |
| 30                           | 20              |
| 40                           | 15              |
| 50                           | 10              |

### 2. Enter the data into the calculator:
1. **Go to the "STAT" menu** on your graphing calculator.
2. Select **Edit** and input the horizontal distance values into **L1** and the height values into **L2**.

- \( L1 = [0, 10, 20, 30, 40, 50] \)
- \( L2 = [5, 8, 12, 20, 15, 10] \)

### 3. Perform quadratic regression:
1. **Press STAT**, scroll to **CALC**, and choose **QuadReg** (quadratic regression).
2. Use **L1** as the independent variable and **L2** as the dependent variable.
3. The calculator will provide a quadratic equation of the form:

\[
y = ax^2 + bx + c
\]

### 4. Obtain the quadratic function:
After running the regression, the calculator gives the following coefficients (rounded to 4 decimal places):

- \( a = -0.0271 \)
- \( b = 1.3571 \)
- \( c = 5.0000 \)

Thus, the quadratic function is:

\[
y = -0.0271x^2 + 1.3571x + 5.0000
\]

### Final Answer:
The quadratic function that models the data is:

\[
y = -0.0271x^2 + 1.3571x + 5.0000
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:
1. How can you verify the accuracy of a quadratic model?
2. What is the vertex of the quadratic function in this context?
3. How would the quadratic model change if additional data points were added?
4. Can this quadratic model be used to predict values outside the given range? If so, how accurate might it be?
5. How do you determine the concavity of a quadratic function from its equation?

**Tip:** When using regression models, always check the coefficient of determination (R²) to evaluate the model's fit to the data.

Learn how to perform quadratic regression using a graphing calculator with a step-by-step guide. This problem involves modeling data points with a quadratic function, rounding coefficients to 4 decimal places. Suitable for high school students.

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