### 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)

### Practice Problem

(x+3)(x+3) = x² + 3x + 3x + 9

x² + 6x + 9

y = x² + 6x + 9

(x+3)(x+3) + 4 = x² + 3x + 3x + 9 + 4

x² + 6x + 13

y = x² + 6x + 13

(x – 3)(x – 3) + 2 = x² – 3x – 3x + 9 + 2

x² – 6x + 11

y = x² – 6x + 11

y = (x – 3)² + 4

y = x² -6x + 9 + 4

y = x² -6x + 13

y = (x – 5 )² + 3

y = x² –10x + 25 + 3

y = x² –10x + 28

### Standard Form to Vertex Form

To convert an equation from standard form to vertex form it is sometimes necessary to be comfortable completing the square

y = (x + 1)²

y = (x + 3)²

y = (x + 3)² + 1

y = (x + 3)² – 1

y = (x + 5)²

(x + 5)² + 2 = (x² + 10x + 25) + 2

y = (x + 5)² + 2

(x + 5)² – 4= (x² + 10x + 25) – 4

y = (x + 5)² – 4

(x + 6)² – 2 = (x² + 12x + 36) – 2

y = (x + 6)² – 2

(x + 7)² – 7 = (x² + 14x + 49) – 9

y = (x + 7)² – 9

(x + 9)² – 10 = (x² + 18x + 81) – 10

y = (x + 9)² – 10

(x – 8)² + 7 = (x² – 16x + 64) + 7

y = (x – 8)² + 7

(x + 9)² + 14 = (x²+ 18x + 81) + 14

y = (x + 9)² + 14

(x – 10)² – 5 = (x² – 20x + 100) – 5

y = (x – 10)² – 5

#### When "a" > 1

2x² + 4x + 5 = 2(x² + 2x) + 5

**2**(x² + 2x + **1**) **–2** + 5

2(x² + 2x + 1) –2 + 5

2(x + 1)² +3

y = 2(x + 1)² + 3

2x² + 4x + 6 = 2(x² + 2x) + 6

**2**(x² + 2x + **1**) **–2** + 6

2(x² + 2x + 1) –2 + 6

2(x + 1)² + 4

y = 2(x + 1)² + 4

3x² + 6x + 8 = 3(x² + 2x) + 8

**3**(x² + 2x + **1**) − **3** + 8

3(x + 1)² + 5

y = 3(x + 1)² + 5