Question

Use a system of linear equations to find the quadratic function f(x) = ax² + bx + c that satisfies the given conditions. Solve the system using matrices. f(-2) = 5, f(1) = -4, f(2) = -15.

a. a = -2, b = 3, c = 1
b. a = 1, b = -2, c = -3
c. a = -1, b = 4, c = -7
d. a = 2, b = -1, c = -6

Answer 1

The quadratic function that satisfies the given conditions is f(x) = -2x² + 3x + 1, so the correct option is a. a = -2, b = 3, c = 1. (option a)

Step-by-step explanation:

To find the quadratic function f(x) = ax² + bx + c that satisfies the given conditions, we can set up a system of three equations using the provided information:

1. f(-2) = -2(-2)² + 3(-2) + 1 = 5

2. f(1) = -2(1)² + 3(1) + 1 = -4

3. f(2) = -2(2)² + 3(2) + 1 = -15

Now, we can represent this system of equations in matrix form AX = B, where A is the coefficient matrix, X is the column matrix containing a, b, c, and B is the column matrix containing the constant terms.

`\[ \begin{bmatrix} 4 & -2 & 1 \\ 1 & 3 & 1 \\ 4 & 2 & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 5 \\ -4 \\ -15 \end{bmatrix} \]`

Using matrix inversion, we can solve for X by multiplying both sides by A^(-1):

`\[ X = A^((-1))B \]`

After performing the calculations, we find a = -2, b = 3, c = 1, which corresponds to option a. Therefore, the correct answer is a. a = -2, b = 3, c = 1.(option a)