Quadratic Functions, Quadratic Expressions, Quadratic Equations

1
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.
2
Quadratic Expressions Factored Form
Examples
1.
2.
3
Factoring quadratic expressions
Given where a, b, c
are integers. Case 1 a 1 Since
4
we have
5
Examples
6
Case 2 where a,
b, c are integers and a ? 1. Since

7
we have
8
Examples
9
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.
10
Methods of Solution
Method 1 Factor then the solutions (roots)
of the equation are
11
Examples
12
Method 2 Use the QUADRATIC FORMULA
The real number solutions (roots) of the
quadratic equation are provided
13
The quadratic formula is often written as
The number is called the
discriminant.
14
The Discriminant
Given the quadratic equation If
15
(1) the roots are
(2) the roots are

(3) no real roots.
16
Examples
17
(No Transcript)
18
(No Transcript)
19
Quadratic Functions
The graph of is a
parabola. The graph looks like if
a gt 0 if a lt 0
20
Key features of the graph

• The maximum or minimum point on the graph is
  called the vertex. The x-coordinate of the
  vertex is

21

• The y-intercept the y-coordinate of the point
  where the graph intersects the y-axis.
• The y-intercept is
• 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

22
Examples
Sketch the graph of
vertex y-intercept x-intercepts
23
(No Transcript)
24
Sketch the graph of Vertex y-intercept
x-intercept(s)
25
(No Transcript)
26
Sketch the graph of Vertex y-intercept
x-intercept(s)
27
(No Transcript)