Chapter 7.5, Problem 25WE

To determine

The equation of the parabola in the form $y-k=a{\left(x-h\right)}^2$ having vertex $\left(4,2\right)$ and $x-$ intercept is $3$ .

Answer to Problem 25WE

The equation of the parabola is $y-2=-2{\left(x-4\right)}^2$ .

Explanation of Solution

Given information: It is given that the vertex of the parabola is $\left(4,2\right)$ and it has

  $x-$ intercept equal to $3$ .

Calculation: Since, the parabola has the vertex $\left(4,2\right)$ therefore satisfy the vertex $\left(4,2\right)$ in the form $y-k=a{\left(x-h\right)}^2$ .

  $\begin{array}{c}y-\left(2\right)=a{\left(x-\left(4\right)\right)}^2\mathrm{....}\left\{\left(h,k\right)\to \left(4,2\right)\right\}\\ y-2=a{\left(x-4\right)}^2\mathrm{....}\left(1\right)\end{array}$

Now the $x-$ intercept of parabola is equal to $3$ . Therefore substitute $x=3\text{ and }y=0$ in (1) because for $x-$ intercept the $y-$ intercept will always be equal to $0$ . Therefore from (1),

  $\begin{array}{c}\left(0\right)-2=a{\left(\left(3\right)-4\right)}^2\mathrm{....}\left\{\left(x,y\right)\to \left(3,0\right)\right\}\\ -2=a\\ \text{or, }a=-2\end{array}$

Again substitute $a=-2$ in (1) gives the equation of parabola. Therefore from (1) equation of parabola is $y-2=-2{\left(x-4\right)}^2$ .