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Notes for IB maths SL - chapter 1, quadratics

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Taken from Haese Mathematics: Mathematics for the international student: Mathematics SL
Chapter 1
Quadratics
Alzbeta Bavorova
Ch1 – Quadratics
=0
The
roots
or
solutions
of
=0
are the values of x for which the equation is true.
Factorization
Quick if you have “easy” numbers.
Tricky if
a =/= 1
. 1.
Rearrange equation to have zero on one side and the rest on the other. 2.
Divide by
a
. 3.
Find numbers
s
and
r
, for which:
a * c = s * r
b = s + r
4.
Rewrite equation as
()()=0
5.
x
1
= -s
;
x
2
= -r
Completing the square
Perfect squares
are expressions such as:
(x + 1)
2
, (x + 2)
2
=> (x + a)
2
Use the formulas
()
=
2
and
()
=
2
41=0
1.
Move
c
to the other side.
4=1
2.
Pick a corresponding new
c
by dividing the
b
by 2 and squaring it.
=(
)
=4
3.
Add c
NEW
to both sides
44= 3
4.
Factorize and solve
(2)
=3
2=±√3
,
= ±√32
Quadratic formula
Foolproof.
,
=±√
42
,
= ±√ ∆2
The discriminant (Δ) of a quadratic
and sign diagrams
∆ =
4
If:
Δ > 0 ……………….. Two real solutions
Δ = 0 ……………….. One solution
Δ < 0
……………….. No real solution
A
sign diagram
(such as the one showed below) shows for what values is the quadratic positive, zero or negative. The points on the number line represent the
roots
of the equation.
Quadratic functions
=
(If we substitute
y
by 0, we have a quadratic equation.) The graph of a quadratic function is a
parabola
, which is one of the
conic sections
(produced by cutting the cone by a plane parallel to the
cone’s slant side).
(http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Parabola_features.svg/2000px-Parabola_features.svg.png)
If
a > 0
, the parabola is
concave/concave up
(like the image). If
a < 0
, the parabola is
convex/concave down
.
Quadratic form, a =/= 0 Facts y = a(x-p)(x-q)
x-intercepts are
p
and
q
axis of symmetry is
=
+
vertex is
(
+
,
+
)
y = a(x-h)
2
touches x-axis at h
Δ = 0
axis of symmetry is
x = h
vertex is
(h, 0)
y = a(x-h)
2
+ k
axis of symmetry is
x = h
vertex is
(h, k)
y = ax
2
+ bx + c
y-intercept is
c
axis of symmetry is
=
−
vertex is
(
−
,
)
x-intercepts can be calculated using
0 = ax
2
+ bx + c
Sketching graphs by completing the square Positive definite and negative definite quadratics
Positive definite
quadratics
are those which are positive for all values of
x
. This means they have no x-intercept and a > 0.
Negative definite quadratics
are the exact opposite.
Finding a quadratic form its graph
The
roots
of the function are its x-intercepts.

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