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