Chapter 7.7, Problem 29WE

To determine

Tofind: thequadratic equation.

Answer to Problem 29WE

The quadratic equation is $f\left(x\right)=-\frac{1}{3}\left(x^2-4x-32\right)$ .

Explanation of Solution

Given information:

vertex $\left(2,12\right)$ and $x$ -intercepts $-4\text{ and 8}$ .

Calculation: to find the quadratic equation, first use the formula of quadratic formula with roots,

$f\left(x\right)=a\left[x^2-\left(\text{sum of roots}\right)x+\left(\text{product of roots}\right)\right]$

Now, substitute and simplify,

$\begin{array}{l}f\left(x\right)=a\left[x^2-\left(-4+8\right)x+\left(\left(-4\right)\left(8\right)\right)\right]\text{ [substitute]}\\ f\left(x\right)=a\left(x^2-4x-32\right)\text{ [simplify]}\end{array}$

Therefore vertex is $\left(2,12\right)$ . Since, the vertex lies on the graph of quadratic equation, so substitute its coordinates in the equation,

$\begin{array}{l}f\left(x\right)=a\left(x^2-4x-32\right)\\ 12=a\left({\left(2\right)}^2-4\left(2\right)-32\right)\text{ [substitute]}\\ 12=a\left(-36\right)\text{ [add and subtract]}\\ a=-\frac{1}{3}\text{ [simplify]}\end{array}$

Thus, the equation is $f\left(x\right)=-\frac{1}{3}\left(x^2-4x-32\right)$ .