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Converting the Equation of Parabola: Vertex and Standard Forms

From Vertex To Standard Form

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'
When a is positive
$$ a < 0 $$ parabola's opens upwards like a 'U'
When a is negative
  • 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 = -b/2a
axis of symmetry
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)

picture from vertex to standard form equation parabola
Parabola 1 has the vertex form equation :
    y = (x + 3)²
  • To rewrite this equation in standard form
    • Expand (x+3)(x+3)
      Answer


2. Change the parabola's equation from vertex form to standard form.
    y = (x + 3)² + 4
    Answer


3. Change the parabola's equation from vertex form to standard form.
    y = (x - 3 )² + 2
    Answer


4. Convert the equation below from vertex form to standard form.
y - 4 = (x - 3 )²
Answer


5. Change the equation of the parabola below into standard form
y - 3 = (x - 5 )²
Answer

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
Answer


What is the vertex form of the parabola whose standard form equation is
2. y = x² + 6x +9
Answer


What is the vertex form of the parabola whose standard form equation is
3. y = x² + 6x + 10
Answer


Convert the equation below from standard to vertex form.
4. y = x² + 6x + 8
Answer


What is the vertex form of the parabola whose standard form equation is
5. y = x² + 10x + 25
Answer


What is the vertex form of the parabola whose standard form equation is
6. y = x² + 10x + 27
Answer


What is the vertex form of the parabola whose standard form equation is
7. y = x² + 10x + 21
Answer


Convert the equation below from standard to vertex form.
8. y = x² + 12x + 34
Answer


What is the vertex form of the parabola whose standard form equation is
9. y = x² + 14x + 40
Answer


Convert the equation below from standard to vertex form.
10. y = x² + 18x + 71
Answer


What is the vertex form of the parabola whose standard form equation is
11. y = x² – 16x + 71
Answer


What is the vertex form of the parabola whose standard form equation is
12. y = x² + 18x + 95
Answer


Convert the equation below from standard to vertex form.
13. y = x² – 20x + 95
Answer


When "a" > 1



Convert the parabola's equation below to vertex form.
14. y = 2x² + 4x + 5
Answer


Complete the square to convert the equation into vertex form.
15. y =2x² + 4x + 6
Answer


Convert the parabola's equation below to vertex form.
16. y = 3x² + 6x + 8
Answer
Practice Problems