# How do you find out the equation of a parabola given the vertex (1 1/2, 3), and it cuts the y axis at 7 1/2?

Apr 30, 2015

In this way:

The equation of the parabolae ($\infty$) given the vertex are:

$y - {y}_{v} = a {\left(x - {x}_{v}\right)}^{2}$.

So:

$y - 3 = a {\left(x - \frac{11}{2}\right)}^{2}$.

If our parabola cuts the y-axis at $7 \frac{1}{2} = \frac{15}{2}$, it means that passes from the point $\left(\frac{15}{2} , 0\right)$.

Let's use this condition to find $a$:

$0 - 3 = a {\left(\frac{15}{2} - \frac{11}{2}\right)}^{2} \Rightarrow - 3 = a {\left(\frac{4}{2}\right)}^{2} \Rightarrow - 3 = 4 a \Rightarrow$

$a = - \frac{3}{4}$.

So:

$y - 3 = - \frac{3}{4} {\left(x - \frac{11}{2}\right)}^{2}$.

And now, if you need, you can do the counts.