# How do you write a quadratic function in vertex form whose has vertex (3/4,2) and passes through point (2,41/8)?

Oct 4, 2017

$y = 2 {\left(x - \frac{3}{4}\right)}^{2} + 2$

#### Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a is}$
$\text{a multiplier}$

$\text{here } \left(h , k\right) = \left(\frac{3}{4} , 2\right)$

$\Rightarrow y = a {\left(x - \frac{3}{4}\right)}^{2} + 2$

$\text{substitute "(2,41/8)" into the equation for a}$

$\Rightarrow \frac{41}{8} = a {\left(\frac{5}{4}\right)}^{2} + 2$

$\Rightarrow \frac{25}{16} a = \frac{25}{8}$

$\Rightarrow a = \frac{25}{8} \times \frac{16}{25} = 2$

$\Rightarrow y = 2 {\left(x - \frac{3}{4}\right)}^{2} + 2 \leftarrow \textcolor{red}{\text{ in vertex form}}$

Oct 4, 2017

The quadratic equation in vertex form is $y = 2 {\left(x - \frac{3}{4}\right)}^{2} + 2$

#### Explanation:

Vertex is at  (3/4,2) ;h=3/4, k=2. The quadratic equation in

vertex form is $y = a \left(x - {h}^{2}\right) + k \mathmr{and} y = a {\left(x - \frac{3}{4}\right)}^{2} + 2$ . The

point $\left(2 , \frac{41}{8}\right)$ satisfies the equation as it is on the parabola.

$\therefore \frac{41}{8} = a {\left(2 - \frac{3}{4}\right)}^{2} + 2 \mathmr{and} a {\left(\frac{5}{4}\right)}^{2} = \frac{41}{8} - 2$ or

$\frac{25}{16} \cdot a = \frac{41 - 16}{8} \mathmr{and} \frac{\cancel{25}}{16} \cdot a = \frac{\cancel{25}}{8} \therefore$

$a = \frac{16}{8} = 2 \therefore$ Hence the quadratic equation in vertex form

is $y = 2 {\left(x - \frac{3}{4}\right)}^{2} + 2$ [Ans]