# How do you write the equation for a graph that is a parabola with a vertex at (5, 3)?

Jul 24, 2017

You can't find any graph with just one point. If you just have a point it's just a point.

#### Explanation:

You can't find the equation for a parabola with just a vertex. You need another point.

The equation for a parabola is $y = a {\left(x - h\right)}^{2} + k$
You have $h = 5$ and $k = 3$ but no $\left(x , y\right)$ so then the equation becomes just $y = a {\left(x - 5\right)}^{2} + 3$ which is not enough information to get the equation.

You can't find any graph with just one point. If you just have a point it's just a point.

Jul 24, 2017

$y = a \left({x}^{2} - 10 x + 25\right) + 3$

#### Explanation:

$y = a {x}^{2} + b x + c$

$- \frac{b}{2 a} = 5 \mathmr{and} - \frac{\Delta}{4 a} = 3$

$b = - 10 a \mathmr{and} \Delta = - 12 a = 100 {a}^{2} - 4 a c \implies c = 25 a + 3$

$y = a {x}^{2} - 10 a x + 25 a + 3$

Example:

${x}^{2} - 10 x + 28 = 0$
$\Delta = 100 - 4 \cdot 28 = - 12$
x = 5 ± i sqrt 3