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> 1, the parabola opens upwards
 if a< 1, it opens downwards.
 The axis of symmetry

The axis of symmetry is the line x = b/2a
 The role of 'a'
Vertex Form of Equation
The vertex form of a parabola's equation is generally expressed as :
y= a(xh)^{2}+k
The vertex form of a parabola's equation is generally expressed as :
y= a(xh)^{2}+k
 (h,k) is the vertex
 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 .
 If a < 1, the graph of the graph becomes narrower(The effect is the opposite of a > 1).
This page: Vertex Form Equation  Graph parabola from Vertex Form  Identify Vertex from Equation
Standard Form Equation
Related Pages: Parabola Home Axis of Symmetry Standard and Vertex Forms of EquationReal World Applications  Converting between Standard and Vertex Forms  Tangent of Parabola area of parabola  pictures
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)²To convert equation to standard form simply expand and simplify the binomial square (Refresher on FOIL to multiply binomials)
Parabola 1 has the vertex form equation :
5. Change the equation of the parabola below into standard form

y = (x + 3)²
 To rewrite this equation in standard form
 Expand (x+3)(x+3)
Answer
(x+3)(x+3) = x² + 3x + 3x + 9
x² + 6x + 9y = x² + 6x + 9
 Expand (x+3)(x+3)

y = (x + 3)² + 4
Answer 
(x+3)(x+3) + 4 = x² + 3x + 3x + 9 + 4
x² + 6x + 13
y = x² + 6x + 13

y = (x  3 )² + 2
Answer 
(x – 3)(x – 3) + 2 = x² – 3x – 3x + 9 + 2
x² – 6x + 11
y = x² – 6x + 11
y  4 = (x  3 )²
Answer 
y = x² 6x + 9 + 4
y = x² 6x + 13
5. Change the equation of the parabola below into standard form
y  3 = (x  5 )²
Answer 
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.
Convert the equation below from standard to vertex form.
1. y = x² + 2x + 1
y = (x + 1)²
Answer
What is the vertex form of the parabola whose standard form equation is
2. y = x² + 6x +9
y = (x + 3)²
Answer
What is the vertex form of the parabola whose standard form equation is
3. y = x² + 6x + 10
y = (x + 3)² + 1
Answer
Convert the equation below from standard to vertex form.
4. y = x² + 6x + 8
y = (x + 3)² – 1
Answer
What is the vertex form of the parabola whose standard form equation is
5. y = x² + 10x + 25
y = (x + 5)²
Answer
What is the vertex form of the parabola whose standard form equation is
6. y = x² + 10x + 27
(x + 5)² + 2 = (x² + 10x + 25) + 2
Answer
y = (x + 5)² + 2
What is the vertex form of the parabola whose standard form equation is
7. y = x² + 10x + 21
(x + 5)² – 4= (x² + 10x + 25) – 4
Answer
y = (x + 5)² – 4
Convert the equation below from standard to vertex form.
8. y = x² + 12x + 34
(x + 6)² – 2 = (x² + 12x + 36) – 2
Answer
y = (x + 6)² – 2
What is the vertex form of the parabola whose standard form equation is
9. y = x² + 14x + 40
(x + 7)² – 7 = (x² + 14x + 49) – 9
Answer
y = (x + 7)² – 9
Convert the equation below from standard to vertex form.
10. y = x² + 18x + 71
(x + 9)² – 10 = (x² + 18x + 81) – 10
Answer
y = (x + 9)² – 10
What is the vertex form of the parabola whose standard form equation is
11. y = x² – 16x + 71
(x – 8)² + 7 = (x² – 16x + 64) + 7
Answer
y = (x – 8)² + 7
What is the vertex form of the parabola whose standard form equation is
12. y = x² + 18x + 95
(x + 9)² + 14 = (x²+ 18x + 81) + 14
Answer
y = (x + 9)² + 14
Convert the equation below from standard to vertex form.
13. y = x² – 20x + 95
(x – 10)² – 5 = (x² – 20x + 100) – 5
Answer
y = (x – 10)² – 5
When "a" > 1
Convert the parabola's equation below to vertex form.
14. y = 2x² + 4x + 5
2x² + 4x + 5 = 2(x² + 2x) + 5
Answer
2(x² + 2x + 1) –2 + 5
2(x² + 2x + 1) –2 + 5
2(x + 1)² +3
y = 2(x + 1)² + 3
Complete the square to convert the equation into vertex form.
15. y =2x² + 4x + 6
2x² + 4x + 6 = 2(x² + 2x) + 6
Answer
2(x² + 2x + 1) –2 + 6
2(x² + 2x + 1) –2 + 6
2(x + 1)² + 4
y = 2(x + 1)² + 4
Convert the parabola's equation below to vertex form.
16. y = 3x² + 6x + 8
3x² + 6x + 8 = 3(x² + 2x) + 8
Answer
3(x² + 2x + 1) − 3 + 8
3(x + 1)² + 5
y = 3(x + 1)² + 5
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