Use technology to find the quadratic regression curve through the given points.(Round all coefficients to four decimal places.)
(1, 4), (3, 6), (4, 5), (5, 3)
y(x) = 
 


To find the quadratic regression curve \( y(x) = ax^2 + bx + c \) for the given points (1, 4), (3, 6), (4, 5), and (5, 3), we can use a quadratic regression method, typically solved using a matrix or statistical tool. Let's proceed step-by-step:

The quadratic regression equation is:

\[
y(x) = ax^2 + bx + c
\]

We need to solve for the coefficients \( a \), \( b \), and \( c \) that minimize the error between the quadratic model and the given points.

### 1. Set up the system of equations using the points:
For each point \( (x, y) \), substitute into the quadratic equation \( y = ax^2 + bx + c \).

- For \( (1, 4) \): \( 4 = a(1)^2 + b(1) + c \) → \( a + b + c = 4 \)
- For \( (3, 6) \): \( 6 = a(3)^2 + b(3) + c \) → \( 9a + 3b + c = 6 \)
- For \( (4, 5) \): \( 5 = a(4)^2 + b(4) + c \) → \( 16a + 4b + c = 5 \)
- For \( (5, 3) \): \( 3 = a(5)^2 + b(5) + c \) → \( 25a + 5b + c = 3 \)

### 2. Solve the system of equations:
We now solve the following system of equations:

1. \( a + b + c = 4 \)
2. \( 9a + 3b + c = 6 \)
3. \( 16a + 4b + c = 5 \)
4. \( 25a + 5b + c = 3 \)

I'll calculate the values of \( a \), \( b \), and \( c \).

It seems I overestimated the number of points required for a quadratic regression (3 points would suffice). However, let's use a more appropriate method to perform quadratic regression for these points, and I'll determine the correct coefficients using a different approach.

Let me try a different calculation method.The quadratic regression equation that best fits the given points is:

\[
y(x) = -0.6136x^2 + 3.4227x + 1.2000
\]

Would you like further details or have any questions?

Here are five related questions to expand on this concept:
1. How is quadratic regression different from linear regression?
2. Can you use this method to predict values for \(x\) beyond the given points?
3. How can technology like graphing calculators or software be used to check regression results?
4. How do you interpret the coefficients in a quadratic regression equation?
5. What are the conditions under which a quadratic regression model is appropriate?

**Tip:** You can use quadratic regression to model data that forms a parabolic shape, indicating the presence of a maximum or minimum.

Use technology to find the quadratic regression curve through the given points. (Round all coefficients to four decimal places.) (1, 4), (3, 6), (4, 5), (5, 3) y(x) =

Learn how to find the quadratic regression curve for the points (1, 4), (3, 6), (4, 5), and (5, 3). This problem covers key concepts in quadratic regression and curve fitting, utilizing the equation y(x) = ax^2 + bx + c. Suitable for students in Grades 10-12 or introductory college level.

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