# What is the equation of the parabola that has a vertex at  (38, -22)  and passes through point  (3,7) ?

Jul 17, 2017

There are 2 standard equations:

$y = \frac{29}{1225} {\left(x - 38\right)}^{2} - 22$

$x = - \frac{35}{841} {\left(y - \left(- 22\right)\right)}^{2} + 38$

#### Explanation:

Actually, there are two such parabolas, one whose general vertex form is:

$y = a {\left(x - h\right)}^{2} + k \text{ [1]}$

and the other whose general vertex form is:

$x = a {\left(y - k\right)}^{2} + h \text{ [2]}$

where $\left(h , k\right)$ is the vertex and "a" is found by substituting in the given point $\left(x , y\right)$

Substitute the vertex $\left(38 , - 22\right)$ into equations [1] and [2]:

$y = a {\left(x - 38\right)}^{2} - 22 \text{ [1.1]}$

$x = a {\left(y - \left(- 22\right)\right)}^{2} + 38 \text{ [2.1]}$

Substitute the point $\left(3 , 7\right)$ into equations [1.2] and [2.1] and then solve for "a":

$7 = a {\left(3 - 38\right)}^{2} - 22$ and $3 = a {\left(7 - \left(- 22\right)\right)}^{2} + 38$

$29 = a {\left(3 - 38\right)}^{2}$ and $- 35 = a {\left(7 - \left(- 22\right)\right)}^{2}$

$29 = a {\left(- 35\right)}^{2}$ and $- 35 = a {\left(29\right)}^{2}$

$a = \frac{29}{1225}$ and $a = - \frac{35}{841}$

Substitute the values for "a" into its respective equation:

$y = \frac{29}{1225} {\left(x - 38\right)}^{2} - 22 \text{ [1.2]}$

$x = - \frac{35}{841} {\left(y - \left(- 22\right)\right)}^{2} + 38 \text{ [2.2]}$

Here is a graph of the two points and both parabolas: