Download - Quadratic Functions
Quadratic Functions, Quadratic Expressions, Quadratic Equations
Definition: A quadratic function is a function of the form
where a, b, c are real numbers and a 0.
The expression on the right-hand-side
is call a quadratic expression.
cbxaxxf 2)(
cbxax 2
Quadratic Expressions: Factored Form
Examples:
form factored exp. quadratic
)3)(12(352
form factored exp. quadratic
)3)(2(65
2
2
xxxx
xxxx
2.
1.
Factoring quadratic expressions:
Given where a, b, c are integers.
Case 1: a = 1; Since
pqxqpx
pqpxqxx
qxpqxxqxpx
)(
)()())((
2
2
2 cbxax
cbxx 2
we have
pqcqpb
qxpxcbxx
and
ifonly and if
))((2
Examples:
56
127
152
86
2
2
2
2
xx
xx
xx
xx
Case 2: where a, b, c areintegers and a 1.
Since
cbxax 2
pqxpsrqrsx
pqpsxrqxrsxqsxprx
)(
))((2
2
we have
cpqbpsrqars
qsxprxcbxax
,,
ifonly and if
))((2
Examples:
376
5163
352
2
2
2
xx
xx
xx
Quadratic Equations:
A quadratic equation is an equation of the form:
Problem: Find the real numbers x, if any, that satisfy the equation.
The numbers that satisfy the equation are called solutions or roots.
02 cbxax
Methods of Solution:
02 cbxax
Method 1: Factor
then the solutions (roots) of the equationare
cbxax 2
))((If 2 qsxprxcbxax
s
qx
r
px 21 and
Examples:2
2
2 15 0
( 5)( 3) 0
5, 3
2 7 3 0
(2 1)( 3) 0
13,
2
x x
x x
x x
x x
x x
x x
The real number solutions (roots) of the quadratic equation
are:
provided
2 0ax bx c
2 2
1 2
4 4,
2 2
b b ac b b acx x
a a
2 4 0.b ac
Method 2: Use the QUADRATIC FORMULA
The quadratic formula is often written as
2 4
2
b b acx
a
The number is called the discriminant.
2 4b ac
The Discriminant:
Given the quadratic equation
If:
2 0.ax bx c
2
2
2
4 0, the equation has real, unequal roots
4 0, the equation has real, equal roots
4 0, the equation has complex roots
b ac
b ac
b ac
2 4 0b ac (1) ; the roots are:
2 2
1 2
4 4;
2 2
b b ac b b acx x
a a
(2) the roots are: 2 4 0;b ac
1 2 2
bx x
a
(3) no real roots.2 4 0;b ac
Examples:
2
2
1 2
2 7 3 0
( 7) ( 7) 4(2)(3)
2(2)
7 49 24 7 25 7 5
4 4 47 5 7 5 1
3,4 4 2
x x
x
x x
2
2
1 2
3 5 0
3 3 4(1)( 5)
2(1)
3 29
2
3 29 3 29,
2 2
x x
x
x x
2
2 2
3 4 4 0
4 ( 4) 4(3)(4) 16 48 32 0
No real roots.
x x
b ac
Quadratic Functions:
The graph of is a parabola. The graph looks like
if a > 0 if a < 0
2( )f x ax bx c
Key features of the graph:
1. The maximum or minimum point on the graph is called the vertex. The x-coordinate of the vertex is:
2
bx
a
( / 2 ) is the -coordinate of the vertex. f b a y
2. The y-intercept; the y-coordinate of the point where the graph intersects the y-axis.
The y-intercept is:
3. The x-intercepts; the x-coordinates of the points, if any, where the graph intersects the x-axis. To find the x-intercepts, solve the quadratic equation
(0) .f c
2 0.ax bx c
Examples:
Sketch the graph of 2( ) 2 8.f x x x
vertex:
y-intercept:
x-intercepts:
21, (1) 9; vertex (1, 9)
2 2
bx f
a
(0) 8f
2 2 8 ( 4)( 2) 0
4, 2.
x x x x
x x
Sketch the graph of
Vertex:
y-intercept:
x-intercept(s):
2( ) 4 4f x x x
42, (2) 0; (2,0)
2( 1)x f
(0) 4f
2
2
4 4 0
4 4 0; ( 2) 0; 2
x x
x x x x
Sketch the graph of
Vertex:
y-intercept:
x-intercept(s):
2( ) 4 5f x x x
42, (2) 1; (2,1)
2(1)x f
(0) 5f
2 4 5 0 has no real solutions.x x