# How do you determine the vertex and direction when given a quadratic function?

Nov 24, 2014

There are two forms a quadratic function could be written in: standard or vertex form. Here are the following ways you can determine the vertex and direction dependent on the form:

Standard Form ($f \left(x\right) = a {x}^{2} + b x + c$)
1. Direction of the parabola can be determined by the value of a. If a is positive, then the parabola faces up (making a u shaped). If a is negative, then the parabola faces down (upside down u).
2. Vertex can be found by $x = - \frac{b}{2 a}$ and then plugging in that value to find y.
Here is an example:
$y = - 2 {x}^{2} + 4 x - 3$, Faces downward since a = -2.
to find the vertex: $x = \frac{- 4}{2 \left(- 2\right)} = \frac{- 4}{- 4} = 1$
then plug that value into the equation$y = - 2 {x}^{2} + 4 x - 3$
$y = - 2 {\left(1\right)}^{2} + 4 \left(1\right) - 3$
$y = - 2 \left(1\right) + 4 \left(1\right) - 3$
$y = - 2 + 4 - 3$
$y = - 1$
Vertex is (1,-1)

Vertex Form ($y = a {\left(x - h\right)}^{2} + k$)
1. Direction of the parabola can be determined by the value of a. If a is positive, then the parabola faces up (making a u shaped). If a is negative, then the parabola faces down (upside down u).
2. Vertex is (h,k).
Here is an example:
$y = - 3 \left(x - 2\right) + 6$ Faces down since a = -3 and the vertex is (2, 6).