1. Find the function f (x) =ax2 + bx+ c whose graph contains the points (1, 2), (-2,-7), and (2, -3). Write the system of equations you obtain in matrix form. Put the matrix in row echelon form and solve for a, b, and c. Classify the system of equations as consistent or inconsistent and classify the equations as dependent or independent.

1. The function is -2x^2 + x + 3.

Step-by-step explanation:

1. The point at (1,2) gives the equation:

a * (1^2) + 1b + c = 2

a + b + c = 2

The point at (-2,-7) gives the equation:

a * (-2^2) + (-2b) + c = 7

4a - 2b + c = -7

The point at (2,-3) gives the equation:

a * (2^2) + 2b + c = -3

4a + 2b + c = -3

Hence, all three equations are:

a + b + c = 2  

4a - 2b + c = -7  

4a + 2b + c = -3

Now, subtract the second equation from the third equation to get:

(4a + 2b + c) - (4a - 2b + c) = -3 - (-7)

4a - 4a + 2b + 2b + c - c = -3 + 7

4b = 4

b = 1

Hence, let's now update our previous equations:

a + 1 + c = 2         =>         a + c = 1

4a - 2(1) + c = -7         =>         4a + c = -5

4a + 2(1) + c = -3         =>         4a + c = -5

This can be further whittled down into two equations:

a + c = 1

4a + c = -5

Hence, we can now subtract the first equation from the second to get:

(4a + c) - (a + c) = -5 - 1

4a - a + c - c = -6

3a = -6

a = -2

Therefore, c = 3 (a + c = 1, -2 + c = 1, c = 1 + 2, c = 3).

And the function is -2x^2 + x + 3.

Hope this helped!

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