Chapter 9, Problem 6CLT

To determine

To find:The equation of the parabola $y=a{x}^2+bx+c$ passing through the points $\left(0,6\right)$ , $\left(2,3\right)$ and $\left(4,2\right)$ .

Answer to Problem 6CLT

The equation of the parabola is $4y=x^2-8x+24$ .

Explanation of Solution

To obtain the equation of the parabola, substitute each of the three points in the equation $y=a{x}^2+bx+c$ , one by one, to obtain 3 equations in 3 unknowns.

For point $\left(0,6\right)$ ,

  $\begin{array}{c}6=a{\left(0\right)}^2+b\left(0\right)+c\\ c=6\end{array}$  ...(1)

For point $\left(2,3\right)$ ,

  $\begin{array}{c}3=a{\left(2\right)}^2+b\left(2\right)+6\\ 4a+2b=-3\\ 4a=-3-2b\end{array}$  ...(2)

For point $\left(4,2\right)$ ,

  $\begin{array}{c}2=a{\left(4\right)}^2+b\left(4\right)+6\\ 16a+4b=-4\\ 4a+b=-1\end{array}$  ...(3)

Substitute the value of $4a$ from equation (2) in equation (3) to obtain the value of $b$ .

  $\begin{array}{c}\left(-3-2b\right)+b=-1\\ -3-2b+b=-1\\ -b=2\\ b=-2\end{array}$

Substitute the value of $b$ in equation (3) to obtain the value of $a$ .

  $\begin{array}{c}4a+\left(-2\right)=-1\\ 4a-2=-1\\ a=\frac{1}{4}\end{array}$

So, the equation of the parabola is,

  $\begin{array}{c}y=a{x}^2+bx+c\\ =\frac{x^2}{4}-2x+6\\ 4y=x^2-8x+24\end{array}$