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
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
- 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).
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)
Parabola 1 has the vertex form equation :
y = (x + 3)²
- To rewrite this equation in standard form
- Expand (x+3)(x+3)
(x+3)(x+3) = x² + 3x + 3x + 9
x² + 6x + 9
y = x² + 6x + 9
2. Change the parabola's equation from vertex form to standard form.
y = (x + 3)² + 4
(x+3)(x+3) + 4 = x² + 3x + 3x + 9 + 4
x² + 6x + 13
y = x² + 6x + 13
3. Change the parabola's equation from vertex form to standard form.
y = (x - 3 )² + 2
(x – 3)(x – 3) + 2 = x² – 3x – 3x + 9 + 2
x² – 6x + 11
y = x² – 6x + 11
4. Convert the equation below from vertex form to standard form.
y - 4 = (x - 3 )²
y = (x – 3)² + 4
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
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.
Convert the equation below from standard to vertex form.
1. y = x² + 2x + 1
What is the vertex form of the parabola whose standard form equation is
2. y = x² + 6x +9
What is the vertex form of the parabola whose standard form equation is
3. y = x² + 6x + 10
Convert the equation below from standard to vertex form.
4. y = x² + 6x + 8
What is the vertex form of the parabola whose standard form equation is
5. y = x² + 10x + 25
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
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
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
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
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
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
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
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
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
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
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
3(x² + 2x + 1) − 3 + 8
3(x + 1)² + 5
y = 3(x + 1)² + 5
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