Chapter 2: QuadraticFunctions
Syllabus
2.1 Quadratic Equations & Inequalities
Solving quadratic equations by:
○
Completing the square & Quadratic formula
○
Forming quadratic equations from given roots
○
Solving quadratic inequalities
2.2 Types of roots of Quadratic equations
Types of roots & value of discriminant
○
Solving problems
○
2.3 Quadratic Functions
○
Various forms of quadratic function
○
General form:
Effect of change of a, b, c & position of graph
-
Relating position of graph & types of roots
-
Vertex form:
○
Effect of change of a, h, k & position of graph
○
Sketching graphs
○
Solving problems
2.1 Quadratic Equations & Inequalities
•
Solving quadratic equations by: Completing the square & Quadratic formula
Quadratic equation:
ax2 + bx + c = 0
Roots:
x-intercepts of graph
y = ax2 + bx + c
Completing the square:
! "
! "
Formula:
ax2 + bx + !"" - !"" + c = 0
1.
2.
•
ax2 + bx + c = 0 (a = 1)
x2 + bx
=-c
"
3.
x2 + bx + !− !"# = - c + !!"#
4.
!𝑥 − # = - c + ! #
5.
𝑥 − " = ± '−𝑐 + !"#
! "
"
𝑥=
−𝑏 ± √𝑏 ! − 4𝑎𝑐
2𝑎
"
! "
"
! "
!
Forming quadratic equations from given roots
𝑥 " − (𝛼 + 𝛽)𝑥 + (𝛼𝛽) = 0
SOR:
𝛼+𝛽 =−
•
!
$
POR:
𝛼𝛽 =
%
$
Solving quadratic inequalities
Method:
▪
Graph sketching
▪
Number line
𝑎" − 𝑏 " = (𝑎 + 𝑏)(𝑎 − 𝑏)
(𝑥 − 𝑦)" = 𝑥 " − 2𝑥𝑦 + 𝑦 " ⇢ 𝑥 " + 𝑦 " = (𝑥 − 𝑦)" + 2𝑥𝑦
(𝑥 + 𝑦)" = 𝑥 " + 2𝑥𝑦 + 𝑦 " ⇢ 𝑥 " + 𝑦 " = (𝑥 + 𝑦)" − 2𝑥𝑦
•
Solving quadratic inequalities
Method:
▪
Graph sketching
▪
Number line
▪
Table
2.2 Types of roots of Quadratic equations
•
Types of roots & value of discriminant
Discriminant, D :
•
b2 - 4ac
Solving problems
Applying
👈
Types of Roots
b2 - 4ac > 0 : 2 real and distinct roots
b2 - 4ac = 0 : 2 real and equal roots
b2 - 4ac < 0 : no real roots
2.3 Quadratic Functions
•
Various forms of quadratic function
General form :
𝑓(𝑥) = 𝑎𝑥 " + 𝑏𝑥 + 𝑐
Vertex form :
𝑓(𝑥) = 𝑎(𝑥 − ℎ)" + 𝑘
Intercept form : 𝑓 (𝑥 ) = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)
Expansion
Factorisation
General
Vertex
Completing the square
•
Intercept
Expansion
General Form:
𝑓(𝑥) = 𝑎𝑥 " + 𝑏𝑥 + 𝑐
-
Effect of change of a, b, c & position of graph
a : shape of graph
-
a > 0 : concave upward
-
a < 0 : concave downward
-
a ↑ , width of curve ↓
(U)
(n)
-
Relating position of graph & types of roots
b2 - 4ac > 0 : intersect x-axis at 2 points
b2 - 4ac = 0 : touches x-axis at 1 point only
b2 - 4ac < 0 : does not intersect at any point on
x-axis
b : axis of symmetry
!
$! %$"
- 𝑥="#
"
b = 0 , axis of symmetry = y-axis
-
a > 0 , b > 0 : left
-
a < 0 , b > 0 : right
b < 0 : right
b < 0 : left
c : y-intercept
•
Vertex Form:
𝑓(𝑥) = 𝑎(𝑥 − ℎ)" + 𝑘
►
Intercept Form
𝑓(𝑥 ) = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)
c : y-intercept
•
Vertex Form:
►
Intercept Form
𝑓(𝑥) = 𝑎(𝑥 − ℎ)" + 𝑘
-
𝑓(𝑥 ) = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)
Effect of change of a, h, k & position of graph
a : shape of graph
•
-
a > 0 : concave upward
(U)
-
a < 0 : concave downward
(n)
a : shape of graph
-
a > 0 : concave upward
(U)
-
a < 0 : concave downward
(n)
h : axis of symmetry
p,q : x-intercept
k : minimum/ maximum value
Axis of symmetry:
-
a > 0 : minimum value
-
a < 0 : maximum value
Sketching graphs
Steps:
•
1.
x-axis & y-axis.
[label]
2.
Graph shape
[a]
Applying
3.
y-intercept
[c/ f(0)=y]
4.
x-intercept
[f(x)=0]
5.
min / max point
[𝑥 =
-
!
"#
]
Making generalisation:
1.
Types of roots
2.
Shape of graph:
parabola, concave …
3.
x-intercept:
Intersect x-axis at …
4.
Solving problems
Max/min point:
Pass through …
Example 1:
Solve the quadratic equation by using completing the square method.
4𝑥 " − 3𝑥 − 2 = 0
👆
𝑥=
&%'
"