Solving Quadratics - PowerPoint PPT Presentation

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Solving Quadratics

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Solving Quadratics Quadratic Formula Discriminant Nature of the Roots Let s explore some more .we might need to revise our generalization. – PowerPoint PPT presentation

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Title: Solving Quadratics


1
Solving Quadratics
  • Quadratic Formula
  • Discriminant Nature of the Roots

2
Warm Up Question 1
  • Given
  • and
  • find an expression for the composite function
  • and state its domain
    restrictions.

3
Warm Up Answer
4
Warm Up Question 2
  • Given
  • a)
  • b)

5
(No Transcript)
6
(No Transcript)
7
Solve by Quadratic Formula
8
Derive the Quadratic Formulaby Completing the
Square
9
Solve using the Quadratic Formula
10
Solve using the Quadratic Formula
11
Solve using the Quadratic Formula
12
Solve using the Quadratic Formula
13
Solve using the Quadratic Formula
14
Solve using the Quadratic Formula
15
Solve using the Quadratic Formula
16
Solve using the Quadratic Formula
17
The Discriminant
  • The nature of the roots

18
The Discriminant
  • It comes from the quadratic formula.

19
  • When you apply the quadratic formula to any
    quadratic equation, you will find that the value
    of b²-4ac is either positive, negative, or 0.
  • The value of the discriminant is what tells us
    the nature of the roots (solutions) to the
    quadratic.

20
Solutions of a Quadratic Equation
  • If

21
Real Numbers (R)
Rational Numbers (Q)
Irrational Numbers
Integers (Z)
Decimal form is non-terminating and non-repeating
Whole Numbers
Natural Numbers (N)
1, 2, 3,
0, 1, 2, 3,
-3, -2, -1, 0, 1, 2, 3,
Decimal form either terminates or repeats
All rational and irrational numbers
22
Find the Discriminant and Describe its Roots.
  • Nature of the Roots

23
Find the Discriminant and Describe its Roots.
  • Nature of the Roots

24
Find the Discriminant and Describe its Roots
  • Nature of the Roots

25
Graphs of Polynomial Functions
26
Explore Look at the relationship between the
degree sign of the leading coefficient and the
right- and left-hand behavior of the graph of the
function.
27
Explore Look at the relationship between the
degree sign of the leading coefficient and the
right- and left-hand behavior of the graph of the
function.
28
Explore Look at the relationship between the
degree sign of the leading coefficient and the
right- and left-hand behavior of the graph of the
function.
29
Continuous Function
  • A function is continuous if its graph can be
    drawn with a pencil without lifting the pencil
    from the paper.

Continuous
Not Continuous
30
Polynomial Function
  • Polynomial Functions have continuous graphs with
    smooth rounded turns.
  • Written
  • Example

31
Explore using graphing CalculatorDescribe graph
as S or W shaped.
Function Degree of U turns
     
     
     
     
     
     
     
     
32
Generalizations?
  • The number of turns is one less than the degree.
  • Even degree ? W Shape
  • Odd degree ? S Shape

33
Describe the Shape and Number of Turns.
34
Lets explore some more.we might need to revise
our generalization.
  • Take a look at the following graph and tell me if
    your conjecture is correct.

35
Lead Coefficient Test
When n is odd
Lead Coefficient is Positive (an gt0), the graph
falls to the left and rises to the right
Lead Coefficient is Negative (an lt0), the graph
rises to the left and falls to the right
36
Lead Coefficient Test
When n is even
Lead Coefficient is Positive (an gt0), the graph
rises to the left and rises to the right
Lead Coefficient is Negative (an lt0), the graph
falls to the left and falls to the right
37
Leading Coefficient an
End Behavior - End Behavior - End Behavior - End Behavior - End Behavior -
         
  left right left right
n - even        
n - odd        
a gt 0
a lt 0
38
Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
39
Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
40
Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
41
Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
42
A polynomial function (f) of degree n , the
following are true
  • The function has at most n real zeros
  • The graph has at most (n-1) relative extrema
    (relative max/min)

43
Local Max / Min (in terms of y)Increasing /
Decreasing (in terms of x)
44
Local Max / Min (in terms of y)Increasing /
Decreasing (in terms of x)
45
Approximate any local maxima or minima to the
nearest tenth.Find the intervals over which the
function is increasing and decreasing.
46
Find the Zeros of the polynomial function below
and sketch on the graph
47
Find the Zeros of the polynomial function below
and sketch on the graph
Multiplicity of 2 EVEN - Touches
48
Find the Zeros of the polynomial function below
and sketch on the graph
49
Find the Zeros of the polynomial function below
and sketch on the graph
NO X-INTERCEPTS!
50
Find the Zeros of the polynomial function below
and sketch on the graph
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