How do you determine the equation of the parabola with vertex (2,6) and passes through (7,2)?

Jul 27, 2016

$y = - \frac{4}{25} {\left(x - 2\right)}^{2} + 6$

Explanation:

The equation of a parabola in $\textcolor{b l u e}{\text{vertex form}}$ is

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where (h ,k) are the coordinates of the vertex and a, a constant.

here (h ,k) = (2 ,6)

$\Rightarrow y = a {\left(x - 2\right)}^{2} + 6 \text{ is the partial equation}$

Given that the parabola passes through (7 ,2) then the coordinates of this point will satisfy the equation.
Substituting x = 7 and y = 2 into the equation allows the constant a to be found.

$\Rightarrow a {\left(7 - 2\right)}^{2} + 6 = 2 \Rightarrow 25 a = - 4 \Rightarrow a = - \frac{4}{25}$

$\Rightarrow y = - \frac{4}{25} {\left(x - 2\right)}^{2} + 6 \text{ is the equation in vertex form}$