Use a system of equations to find the quadratic function f(x) = ax^2 + bx + c
that satisfies the equations. Solve the system using matrices.
f(−2) = 4, f(1) = −2, f(2) = −12
f(x) = ax^2 + bx + c
form the equations, using the 3 given points.
4a - 2b + c = 4
a + b + c = -2
4a + 2b + c = -12
These are too easy to complicate things with matrices
subtract the first two:
3a -3b = 6
a - b = 2
subtract the last two
3a + b = -10
add these two :
4a = -8
a = -2
sub back into 3a+b=-10 to get b
then sub back into a + b + c = -2 to get your c
To find the quadratic function that satisfies the given equations, we can set up a system of three equations with three variables: a, b, and c. We'll use the values for x from the given equations to form the system.
Step 1: Set up the equations using the given values:
Since the quadratic function is f(x) = ax^2 + bx + c, we substitute the x-values from the given equations:
1. (-2)^2*a + (-2)b + c = 4
2. (1)^2*a + (1)b + c = -2
3. (2)^2*a + (2)b + c = -12
Step 2: Write the system of equations in matrix form:
We can rewrite the system of equations in matrix form as follows:
⎡ 4 ⎤ ⎡ (-2)^2 -2 1 ⎤ ⎡ a ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢-2⎥ = ⎢ (1)^2 1 1 ⎥ x ⎢ b ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣-12⎦ ⎣ (2)^2 2 1 ⎦ ⎣ c ⎦
Step 3: Solve the system using matrices:
To solve the system, we need to find the inverse of the coefficient matrix (the matrix on the right-hand side) and multiply it by the constant matrix (the matrix on the left-hand side).
⎡ a ⎤ ⎡ (-2)^2 -2 1 ⎤^-1 ⎡ 4 ⎤
⎢ ⎥ = ⎢ ⎥ ⎢ ⎥
⎢ b ⎥ ⎢ (1)^2 1 1 ⎥ ⎢-2⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ c ⎦ ⎣ (2)^2 2 1 ⎦ ⎣-12⎦
Solving this matrix equation will give us the values of a, b, and c, which represent the coefficients of the quadratic function f(x).
Note: To find the inverse of a matrix, you can either use matrix algebra techniques or use online matrix calculators if you prefer.
To find the quadratic function that satisfies the given equations, we need to write a system of equations using the given information.
The general form of a quadratic function is f(x) = ax^2 + bx + c. Plugging in the given values of x and f(x) into this equation will give us a system of equations in terms of a, b, and c.
We are given three points: (-2, 4), (1, -2), and (2, -12). Plugging in these values into the equation f(x) = ax^2 + bx + c, we get:
For the point (-2, 4): 4 = a(-2)^2 + b(-2) + c
Simplifying this equation, we get: 4 = 4a - 2b + c ... (Equation 1)
For the point (1, -2): -2 = a(1)^2 + b(1) + c
Simplifying this equation, we get: -2 = a + b + c ... (Equation 2)
For the point (2, -12): -12 = a(2)^2 + b(2) + c
Simplifying this equation, we get: -12 = 4a + 2b + c ... (Equation 3)
Now, we have a system of three equations (Equations 1, 2, and 3) in terms of a, b, and c. We can represent this system using matrices and solve for a, b, and c using matrix operations.
First, we can rewrite the equations in the form of AX = B, where A is the coefficients matrix, X is the variables matrix, and B is the constant matrix.
The coefficient matrix (A) will be:
| 4 -2 1 |
| 1 1 1 |
| 4 2 1 |
The variables matrix (X) will be:
| a |
| b |
| c |
The constant matrix (B) will be:
| 4 |
|-2 |
|-12|
To solve for X, we can use the formula X = A^(-1) * B, where A^(-1) is the inverse of matrix A.
Once you find the inverse of matrix A, you can multiply the inverse by matrix B to solve for X, which will give you the values of a, b, and c.
By using a matrix calculator or software that can perform matrix operations, you can find the inverse of matrix A and find the solution for X. Then, plug in the values of a, b, and c in the equation f(x) = ax^2 + bx + c to obtain the quadratic function that satisfies the given equations.