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To solve the problem, we will follow the steps outlined:

### Given:
We have three points \((1, 7)\), \((2, 10)\), and \((3, 11)\) that lie on a quadratic model of the form \( y = ax^2 + bx + c \).

### (a) Construct three equations in terms of \(a\), \(b\), and \(c\).

Substitute each point into the equation \( y = ax^2 + bx + c \):

1. For the point \((1, 7)\):
   \[
   7 = a(1)^2 + b(1) + c \implies a + b + c = 7
   \]

2. For the point \((2, 10)\):
   \[
   10 = a(2)^2 + b(2) + c \implies 4a + 2b + c = 10
   \]

3. For the point \((3, 11)\):
   \[
   11 = a(3)^2 + b(3) + c \implies 9a + 3b + c = 11
   \]

### (b) Use technology to find \(a\), \(b\), and \(c\).

We have the following system of equations:
\[
\begin{aligned}
1. & \quad a + b + c = 7 \\
2. & \quad 4a + 2b + c = 10 \\
3. & \quad 9a + 3b + c = 11
\end{aligned}
\]

To solve this, we can use substitution, elimination, or a matrix method. Let's solve it using a matrix approach:

\[
\begin{bmatrix}
1 & 1 & 1 \\
4 & 2 & 1 \\
9 & 3 & 1
\end{bmatrix}
\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}
=
\begin{bmatrix}
7 \\
10 \\
11
\end{bmatrix}
\]

I'll solve this system now.### (b) Solution for \(a\), \(b\), and \(c\):
The solutions are:
\[
a = -1, \quad b = 6, \quad c = 2
\]

### (c) Determine the model.
Substitute the values of \(a\), \(b\), and \(c\) into the quadratic equation \( y = ax^2 + bx + c \):

\[
y = -x^2 + 6x + 2
\]

This is the quadratic model that fits the given points.

Would you like a detailed explanation of any part of this solution? 

Here are some related questions you might consider:

1. How can you verify that the points lie on the derived model?
2. What is the vertex of the quadratic model \(y = -x^2 + 6x + 2\)?
3. How does the model change if an additional point is given?
4. What is the axis of symmetry for this quadratic function?
5. Can this quadratic function have a maximum or minimum, and where does it occur?

**Tip:** When solving systems of equations, using matrices can simplify the process, especially for larger systems.

The points (1, 7), (2, 10), and (3, 11) lie on a model of the form y = ax^2 + bx + c. 
 a) Construct three equations in terms of a, b, and c. 
 b) Use technology to find a, b, and c. 
 c) Hence determine the model.

Learn how to find the quadratic equation y = ax^2 + bx + c that passes through three points, (1, 7), (2, 10), and (3, 11). This problem explores key algebraic concepts such as solving systems of equations and quadratic functions. Suitable for high school students in Grades 9-12.

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