The equation of a parabola can be expressed in either standard or vertex form as shown in the picture below.

### Standard Form Equation

The standard form of a parabola's equation is generally expressed:

- y = ax
^{2}+ bx + c The role of 'a'- If a> 0, the parabola opens upwards
- if a< 0, it opens downwards.

The axis of symmetry- The axis of symmetry is the line x = -b/2a

**Picture of Standard form equation**

**Axis of Symmetry from Standard Form**

### Vertex Form of Equation

The vertex form of a parabola's equation is generally expressed as: y = a(x-h)^{2}+k

- (h,k) is the vertex as you can see in the picture below
- If a is positive then the parabola opens upwards like a regular "U".
- If a is negative, then the graph opens downwards like an
**upside down**"U".

- If |a| < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways . Explore the way that 'a' works using our interactive parabola grapher.
- If |a| > 1, the graph of the graph becomes narrower(The effect is the opposite of |a| < 1).

**Practice** Problems

#### Vertex and Direction-Vertex Form Equation

##### Part I

The parabola's vertex is the point (1,1).

#### Identifying the vertex in vertex form

The vertex is the point (-3,4)

(3,4) is the vertex.

vertex is (2, –3)

#### Part II

The vertex is (3,4) and it opens upwards since a is positive( it is 2), it opens upwards.

Vertex = (-3,4), and it opens upwards since a is positive.

Vertex = (9, 5) and since a is negative (it is -22), it opens downwards.