# How do you write a quadratic function in vertex form whose graph has the vertex (-4,-2) and point (-3,-1)?

Aug 25, 2017

There are two vertex forms:

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

and

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

Because a function was requested, then equation [2] must be discarded, because it is not a function.

Substitute the given vertex $\left(h , k\right) = \left(- 4 , - 2\right)$ into equation [1]:

$y = a {\left(x - \left(- 4\right)\right)}^{2} - 2 \text{ [3]}$

Substitute $\left(x , y\right) = \left(- 3 , - 1\right)$ into equation [3] and then solve for "a":

$- 1 = a {\left(- 3 - \left(- 4\right)\right)}^{2} - 2$

$- 1 = a {\left(1\right)}^{2} - 2$

$a = 1$

Substitute 1 for "a" into equation [3]:

$y = {\left(x - \left(- 4\right)\right)}^{2} - 2 \leftarrow$ the answer.

Note: One can write ${\left(x - \left(- 4\right)\right)}^{2}$ as ${\left(x + 4\right)}^{2}$ but it is not recommended, because it can lead to errors, when obtaining the value of h.