Chapter 8.1, Problem 98E

To determine

Find the quadratic function $f\left(x\right)=a{x}^2+bx+c$ .

Answer to Problem 98E

  $f\left(x\right)=-2x^2-2x+1$ is the quadratic function.

Explanation of Solution

Given information: use a system of equations to find the quadratic function $f\left(x\right)=a{x}^2+bx+c$ that satisfies the given conditions. Solve the system using matrices.

  $f\left(-2\right)=-3,f\left(1\right)=-3,f\left(2\right)=-11$

Calculation: The quadratic function $f\left(x\right)=a{x}^2+bx+c$

  $f\left(-2\right)=-3$

Substitute $x=-2$

  $\begin{array}{l}a{\left(-2\right)}^2+b\left(-2\right)+c=-3\\ 4a-2b+c=-\mathrm{3............}\left(1\right)\end{array}$

Now,

  $f\left(1\right)=-3$

  $\begin{array}{l}a{\left(1\right)}^2+b\left(1\right)+c=-3\\ a+b+c=-\mathrm{3...........}\left(2\right)\end{array}$

Also, $f\left(2\right)=-11$

  $a{\left(2\right)}^2+b\left(2\right)+c=-11$

  $4a+2b+c=-\mathrm{11..........}\left(3\right)$

Here equation $\left(1\right)$

  $\left(2\right)$ & $\left(3\right)$ represent the system of equations. We can solve the system by using augmented matrix and Gauss-Jordan elimination method.

  $4a-2b+c=-\mathrm{3............}\left(1\right)$

  $a+b+c=-\mathrm{3...........}\left(2\right)$

  $4a+2b+c=-\mathrm{11..........}\left(3\right)$

The augmented matrix of above system of equations

  $\left[\begin{array}{ccccc}4& -2& 1& ⋮& -3\\ 1& 1& 1& ⋮& -3\\ 4& 2& 1& ⋮& -11\end{array}\right]$

Now apply Gauss-Jordan elimination method.

Interchange row one and two,

  $R_1\iff R_2$ $\left[\begin{array}{ccccc}1& 1& 1& ⋮& -3\\ 4& -2& 1& ⋮& -3\\ 4& 2& 1& ⋮& -11\end{array}\right]$

  $\begin{array}{l}-4R_1+R_2\to R_2\\ -4R_1+R_3\to R_3\end{array}$ $\left[\begin{array}{ccccc}1& 1& 1& ⋮& -3\\ 0& -6& -3& ⋮& 9\\ 0& -2& -3& ⋮& 1\end{array}\right]$

  $\begin{array}{l}-R_2+R_1\to R_1\\ -\left(1/6\right)R_2\to R_2\\ 2R_2+R_3\to R_3\end{array}$ $\left[\begin{array}{ccccc}1& 0& 1/2& ⋮& -9/2\\ 0& 1& 1/2& ⋮& 3/2\\ 0& 0& -2& ⋮& 4\end{array}\right]$

  $-1/2R_3\to R_3$ $\left[\begin{array}{ccccc}1& 0& 1/2& ⋮& -9/2\\ 0& 1& 1/2& ⋮& 3/2\\ 0& 0& 1& ⋮& 2\end{array}\right]$

  $\begin{array}{l}-1/2R_3+R_1\to R_1\\ -1/2R_3+R_2\to R_2\end{array}$ $\left[\begin{array}{ccccc}1& 0& 0& ⋮& -2\\ 0& 1& 0& ⋮& -2\\ 0& 0& 1& ⋮& 1\end{array}\right]$

From reduced matrices solution the system of linear equation has variable,

  $\begin{array}{l}a=-2\\ b=-2\\ c=1\end{array}$

Hence $f\left(x\right)=-2x^2-2x+1$ is the quadratic function.