### Standard Form Equation

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

$ y = ax^2 + bx + c $

#### The role of **a** in $$ \color{Red}{a}x^2 + bx + c $$

$$ a > 0 $$ |
parabola's opens upwards like a 'U' |

$$ a < 0 $$ |
parabolas opens downwards like an upside down 'U' |

- If $$|a| < 1 $$, the graph of the parabola's widens. This just means that the "U" shape of parabola stretches out sideways .
- If $$ |a| > 1 $$, the graph of the graph becomes narrower(The effect is the opposite of |a| < 1).

### Axis of Symmetry in Standard Form

The axis of symmetry is the line $$x =\frac{ -b}{2a} $$

### Vertex Form

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

- (h,k) is the vertex
- If a is positive then the parabola opens upwards like a regular "U". (same as standard form)
- If a is negative, then the graph opens downwards like an
**upside down**"U".(same as standard form) - If |a| < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways .
- If |a| > 1, the graph of the graph becomes narrower(The effect is the opposite of |a| > 1).

### From Vertex To Standard Form

Example of how to convert the equation of a parabola from vertex to standard form.

**Equation in vertex form:**y = (x – 1)²

To convert equation to standard form simply expand and simplify the binomial square (Refresher on FOIL to multiply binomials)