Chapter 2.4, Problem 16E

To determine

To find: The equation which represents the parabola given.

Answer to Problem 16E

Option A $y=2(x-2)(x+1)$ and Option C $y=2{(x-0.5)}^2-4.5$ represents the given parabola.

Explanation of Solution

Given information: $x-$ intercepts $-1\text{and}2$ and vertex $(0.5,-4.5)$ .

Intercept form of a parabola is $y=a(x-p)(x-q)$ .

  $x-$ Intercepts $p\text{and}q$ is $-1\text{and}2$ respectively.

By substituting the $x-$ intercepts in the intercept form of equation we get,

  $y=a(x+1)(x-2)$

To find the value of $a$ , substitute the vertex $(0.5,-4.5)$ in the above equation.

  $\begin{array}{c}-4.5=a(0.5+1)(0.5-2)\\ -4.5=a(1.5)(-1.5)\\ -4.5=a(-2.25)\\ a=\frac{-4.5}{-2.25}=2\end{array}$

Substitute $a=2$ in $y=a(x+1)(x-2)$ ,

  $y=2(x-2)(x+1)$

The above equation is the intercept form of the given parabola.

Vertex form of a parabola is $y=a{(x-h)}^2+k$ .

Vertex $(h,k)$ is $(0.5,-4.5)$ .

By substituting the vertex in the vertex form of equation we get,

  $y=a{(x-0.5)}^2-4.5$

To find the value of $a$ , substitute $(-1,0)$ in the above equation.

  $\begin{array}{c}$ 0=a{(-1-0.5)}^2-4.5$$ 0=a{(-1.5)}^2-4.5$0=a(2.25)-4.5\\ 4.5=a(2.25)\\ a=\frac{4.5}{2.25}=2\end{array}$

Substitute $a=2$ in $y=a{(x-0.5)}^2-4.5$ ,

  $y=2{(x-0.5)}^2-4.5$

The above equation is the vertex form of the given parabola.