“Reference Sheet” – Quadratics
General form: ax + bx + c = 0
Also see:
* Identify:
Ex: f(x) = x2 - 9x + 14
a=1
b = -9
c = 14
a=
b=
c=
* Axis of symmetry:
1) x = -b
2a
f(x) = ax2 + bx + c
f(x) = (x + 2) (x – 1)
(trinomial)
(factors)
Axis of symmetry:
x = -(-9)
2(1)
x = 4.5
2) Draw a dotted line on the grid
3) Label the axis of symmetry
* Vertex:
1) Substitute “x” (from your axis of
symmetry) into the original equation
and solve for “y”. Now you have an
ordered pair (the vertex).
2) Plot the point on the grid
3) Label the point/vertex.
Vertex:
y = (4.5) - 9(4.5) = 14
= 20.25 – 40.5 + 14
= -6.25
(4.5, -6.25)
* X-intercept:
1) Factor the equation
X-intercept: (see back for other possible options)
x2 – 9x + 14
Factors of 14: 14 x 1
-14 x -1
(x-2) (x-7)
7x2
-7 x -2
Combined (added together) = 9
x–2=0
x–7=0
+ 2 +2
+7 = +7
x = 2
x = 7
(2,0) and (7.0)
2) Set each factor equal to 0
3) Solve each factor for x
4) Plot points on grid; label
* Y-intercept:
1) Plug a “0” in place of every “x” in
the original equation
2) The “answer” is where the graph
hits the y-axis
3) Plot point on grid; label
Y-intercept:
y = x2 - 9x + 14
y = 0 - 9(0) + 14
y = 14
* Graph & Label
- vertex
- axis of symmetry
- x-intercepts
- y-intercept
- minimum of 5 points
Graph & Label
These should all be done except for the 5th point.
The 5th point can be done easily if you choose a
value for “x” that is equidistance from the axis
of symmetry as the “x” value used was – when
calculating the y-intercept. (i.e. 0 was 4.5 points
away from the axis of symmetry, so 9 would be
4.5 points away from the axis of symmetry in
the opposite direction). y = x – 9x + 14
y = 9 - 9(9) + 14
y = 81 – 81 + 14
y = 14
(9,14)
- fifth point
(0,14)
Other “miscellaneous things” that will make life easier – depending on the format.
Quadratic equation in factored form: f(x) = a (x-r ) (x-r )
r & r are the roots of the equation
EX: f(x) = - ½ (x+6)(x-2)
the roots are: -6 and 2
Quadratic equation in standard form (vertex form): f(x) = a (x-h) + k
h & k are the x and y coordinates of the vertex
EX: f(x) = 1/3 (x+6) - 3
Discriminant:
when:
b
b
b
b
the vertex is: (-6,-3)
– 4ac
- 4ac > 0 (there are 2 roots)
– 4ac = 0 (the 2 roots are =)
- 4ac < 0 (complex conjugates - imaginary)