# What is the equation of the parabola that has a vertex at  (9, -23)  and passes through point  (35,17) ?

Jan 13, 2016

We can solve this using the vertex formula, $y = a {\left(x - h\right)}^{2} + k$

#### Explanation:

The standard format for a parabola is

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

But there is also the vertex formula,

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

Where $\left(h , k\right)$ is the location of the vertex.

So from the question, the equation would be

$y = a {\left(x - 9\right)}^{2} - 23$

To find a, substitute the x and y values given: $\left(35 , 17\right)$ and solve for $a$:

$17 = a {\left(35 - 9\right)}^{2} - 23$

$\frac{17 + 23}{35 - 9} ^ 2 = a$

$a = \frac{40}{26} ^ 2 = \frac{10}{169}$

so the formula, in vertex form, is

$y = \frac{10}{169} {\left(x - 9\right)}^{2} - 23$

To find the standard form, expand the ${\left(x - 9\right)}^{2}$ term, and simplify to
$y = a {x}^{2} + b x + c$ form.

Jan 13, 2016

For problems of this type, use vertex form, y = a${\left(x - p\right)}^{2}$ + q.

#### Explanation:

In vertex form, mentioned above, the vertex's coordinates are (p, q) and a point (x , y) that is on the parabola.

When finding the equation of the parabola, we have to solve for a, which influences the width and the direction of opening of the parabola.

y = a${\left(x - p\right)}^{2}$ + q
17 = a${\left(35 - 9\right)}^{2}$ - 23
17 = 576a - 23
17 + 23 = 576a
$\frac{5}{72}$ = a

So, the equation of the parabola is y = $\frac{5}{72}$${\left(x - 9\right)}^{2}$ - 23.

Hopefully you understand now!