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High School
Algebra
Algebra I
Quadratic function and equation

gamau6v

Answered question

2021-12-05

Use a system of linear equations to find the quadratic function

f (x) = a x^{2} + b x + c

that satisfies the given conditions. Solve the system using matrices.

f (1) = - 1, f (2) = - 4, f (3) = - 11

f (x) = ?

Answer & Explanation

Xyle1991

Beginner2021-12-06Added 15 answers

Step 1
Given: $f (x) = a x^{2} + b x + c$
$f (1) = - 1$ which means value of function is -1 at $x = 1$
$- 1 = a {(1)}^{2} + b (1) + c$
1) $- 1 = a + b + c$
$f (2) = - 4$
$- 4 = a {(2)}^{2} + b (2) + c$
2) $- 4 = 4 a + 2 b + c$
$f (3) = - 11$
$- 11 = a {(3)}^{2} + b (3) + c$
3) $- 11 = 9 a + 3 a + c$
Step 2
Writing coeficient of a, b and c in form of a matrix for all 3 equation, as
$[\begin{matrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{matrix}] [\begin{matrix} a \\ b \\ c \end{matrix}] = [\begin{matrix} - 1 \\ - 4 \\ - 11 \end{matrix}]$
It is of the form $A x = B$
1) $\Rightarrow x = A^{- 1} B$
Where $x = [\begin{matrix} a \\ b \\ c \end{matrix}]$
$A = [\begin{matrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{matrix}]$
$B = [\begin{matrix} - 1 \\ - 4 \\ - 11 \end{matrix}]$
$A^{- 1} = \frac{a d j A}{| A |} = \frac{(cofactor of A)^{'}}{| A |}$

$| A | = [\begin{matrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{matrix}] = 1 (- 1) - 1 (- 5) + 1 (- 6)$
$= - 1 + 5 - 6 = - 2$
$a d j A = [\begin{matrix} - 1 & 2 & - 1 \\ 5 & - 8 & 3 \\ - 6 & 6 & - 2 \end{matrix}]$
Step 3
$\Rightarrow A^{- 1} = \frac{1}{(- 2)} [\begin{matrix} - 1 & 2 & - 1 \\ 5 & - 8 & 3 \\ - 6 & 6 & - 2 \end{matrix}] = [\begin{matrix} \frac{1}{2} & - 1 & \frac{1}{2} \\ - \frac{5}{2} & 4 & - \frac{3}{2} \\ 3 & - 3 & 1 \end{matrix}]$
Put in (1)

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