Age (x)	Median Income (y)
19.5	            12965
29.5	          33132
39.5	          43638
49.5	           45693
59.5	          41477
69.5	           24502 Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income.
The quadratic function of best fit is yequals=enter your response herexsquared2plus+enter your response herexminus−enter your response here.
​(Type integers or decimals rounded to three decimal places as​ needed.)

To find the quadratic function of best fit for the given data (age vs. median income), we can use quadratic regression. The general form of a quadratic equation is:

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

Where:
- \( y \) is the median income
- \( x \) is the age
- \( a \), \( b \), and \( c \) are the constants to be determined.

We will use the method of least squares to fit the quadratic equation to the dataTo find the quadratic function of best fit for the given data (age vs. median income), we can use quadratic regression. The general form of a quadratic equation is:

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

Where:
- \( y \) is the median income
- \( x \) is the age
- \( a \), \( b \), and \( c \) are the constants to be determined.

We will use the method of least squares to fit the quadratic equation to the 

| Age (x) | Median Income (y) |
|---------|-------------------|
| 19.5    | 12965             |
| 29.5    | 33132             |
| 39.5    | 43638             |
| 49.5    | 45693             |
| 59.5    | 41477             |
| 69.5    | 24502             |

I'll calculate the quadratic regression and determine the best-fit values for \( a \), \( b \), and \( c \).The quadratic function of best fit that models the relation between age and median income is:

\[
y = -43.678x^2 + 4129.575x - 50965.007
\]

This equation can be used to estimate the median income based on age.

Would you like more details on the process, or do you have any questions?

Here are five related questions you could explore:
1. How can you use this quadratic model to predict income for ages outside the given range?
2. What is the vertex of this quadratic function, and what does it represent in this context?
3. How would the model change if you included more data points for other ages?
4. How accurate is the quadratic fit, and are there other models that could provide a better fit?
5. How can you use calculus to find the age at which the median income reaches its maximum?

**Tip:** Quadratic functions are often used to model relationships that have a peak or trough (maximum or minimum), such as income over a lifetime.

Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income. The quadratic function of best fit is y equals= enter your response here x squared 2 plus + enter your response here x minus − enter your response here.

Learn how to use quadratic regression to model the relationship between age and median income. This solution involves fitting a quadratic equation to real-world data using the least squares method, covering key concepts in algebra and curve fitting. Suitable for students in Grades 10-12.

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