Solving Quadratics

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Solving Quadratics

• Quadratic Formula
• Discriminant Nature of the Roots

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Warm Up Question 1

• Given
• and
• find an expression for the composite function
• and state its domain
  restrictions.

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Warm Up Answer
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Warm Up Question 2

• Given
• a)
• b)

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(No Transcript)
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(No Transcript)
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Solve by Quadratic Formula
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Derive the Quadratic Formulaby Completing the
Square
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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Solve using the Quadratic Formula
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The Discriminant

• The nature of the roots

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The Discriminant

• It comes from the quadratic formula.

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• 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.

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Solutions of a Quadratic Equation

• If

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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
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Find the Discriminant and Describe its Roots.

• Nature of the Roots

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Find the Discriminant and Describe its Roots.

• Nature of the Roots

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Find the Discriminant and Describe its Roots

• Nature of the Roots

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Graphs of Polynomial Functions
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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.
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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.
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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.
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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
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Polynomial Function

• Polynomial Functions have continuous graphs with
  smooth rounded turns.
• Written
• Example

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Explore using graphing CalculatorDescribe graph
as S or W shaped.
Function Degree of U turns
     
     
     
     
     
     
     
     
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Generalizations?

• The number of turns is one less than the degree.
• Even degree ? W Shape
• Odd degree ? S Shape

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Describe the Shape and Number of Turns.
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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.

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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
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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
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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
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Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
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Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
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Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
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Use the Leading Coeffiicent Test to describe the
right-hand and left-hand behavior of the graph of
each polynomial function
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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)

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