Equation of a

1
Equation of a Parabola
2
The equation for a parabola can come in 2
forms General form f(x) ax2 bx
c Standard form f(x) a(x h)2 k
Each form has its advantages. General form is
simpler and therefore a little easier to work
with when substituting values. Standard form has
the advantage of easily identifying the vertex
the most important point of the parabola.
3
Identifying the Vertex
To determine the vertex from general form, you
can use a formula to determine the x-value and
another one to determine its image. f(x) ax2
bx c ? b2 4ac
To determine the vertex from standard form, you
use the parameters h and k presented in the
equation. f(x) a(x - h)2 k
Vertex (h,k)
4
Identifying the Axis of Symmetry
After determining the vertex, the axis of
symmetry is determined by using the equation x
x-value of the vertex This is true whether
the parabola is presented in general form or
standard form. Therefore, from general form, the
axis of symmetry is
Therefore, from standard form, the axis of
symmetry is
x h
5
Identifying the Zeros
To determine the zeros from general form, you can
use the quadratic formula. f(x) ax2 bx
c ? b2 4ac This formula can yield either 0,
1 or 2 values depending on the value of the
discriminant. These values manifest themselves
as x-intercepts on the graph.
To determine the zeros from standard form, it is
best to transform the equation to general form
and use the method prescribed above. f(x) a(x
- h)2 k
6
Transforming from Standard Form to General Form
An equation in standard form can be transformed
by first using FOIL to multiply the binomial that
is squared. f(x) -2(x - 4)2 8 f(x) -2(x
4)(x 4) 8 -2(x2 4x 4x 16) 8
-2x2 16x - 32 8 f(x) -2x2 16x - 24
a -2 b 16 c -24 ? b2 4ac 162
4(-2)(-24) 256 - 192 64
7
Identifying the Y-intercept
To determine the y-intercept from general form,
you can use parameter c. f(x) ax2 bx c
Vertex (0,c)
To determine the y-intercept from standard form,
you find f(0). f(x) -2(x - 4)2 8 f(0)
-2(0 - 4)2 8 -2(16) 8 -32 8 -24
Vertex (0,-24)
8
Transforming from General Form to Standard Form
Just as it is possible to transform an equation
in standard form to general form, it is possible
to transform it the other way. Before looking at
this process, it is necessary to review some
concepts.
9
To convert from general form to standard, we must
use a technique called Competing the Square.
This refers to creating a perfect square
trinomial and accommodating the changes within an
equation to make that trinomial.
Some examples of perfect square trinomials are
Notice that the sign of the middle term can be
positive or negative. There is a relationship
between the coefficient of the middle term and
the last term
x2 2x 1
x2 - 4x 4
2 ? 1
-4 ? 4
x2 14x 49
14 ? 49
x2 - 20x 100
-20 ? 100
10
The relationship between these terms of the
trinomial is what makes the trinomial a perfect
square trinomial.
We can use this relationship to create a perfect
square trinomial. If we know the value of the
middle term, we can determine the third term.
x2 32x ___
x2 32x 256
x2 7x ____