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
