Solving Quadratics

- Quadratic Formula
- Discriminant Nature of the Roots

Warm Up Question 1

- Given
- and
- find an expression for the composite function
- and state its domain

restrictions.

Warm Up Answer

Warm Up Question 2

- Given
- a)
- b)

(No Transcript)

(No Transcript)

Solve by Quadratic Formula

Derive the Quadratic Formulaby Completing the

Square

Solve using the Quadratic Formula

Solve using the Quadratic Formula

Solve using the Quadratic Formula

Solve using the Quadratic Formula

Solve using the Quadratic Formula

Solve using the Quadratic Formula

Solve using the Quadratic Formula

Solve using the Quadratic Formula

The Discriminant

- The nature of the roots

The Discriminant

- It comes from the quadratic formula.

- 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.

Solutions of a Quadratic Equation

- If

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

Find the Discriminant and Describe its Roots.

- Nature of the Roots

Find the Discriminant and Describe its Roots.

- Nature of the Roots

Find the Discriminant and Describe its Roots

- Nature of the Roots

Graphs of Polynomial Functions

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.

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.

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.

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

Polynomial Function

- Polynomial Functions have continuous graphs with

smooth rounded turns. - Written
- Example

Explore using graphing CalculatorDescribe graph

as S or W shaped.

Function Degree of U turns

Generalizations?

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

Describe the Shape and Number of Turns.

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.

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

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

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

Use the Leading Coeffiicent Test to describe the

right-hand and left-hand behavior of the graph of

each polynomial function

Use the Leading Coeffiicent Test to describe the

right-hand and left-hand behavior of the graph of

each polynomial function

Use the Leading Coeffiicent Test to describe the

right-hand and left-hand behavior of the graph of

each polynomial function

Use the Leading Coeffiicent Test to describe the

right-hand and left-hand behavior of the graph of

each polynomial function

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)

Local Max / Min (in terms of y)Increasing /

Decreasing (in terms of x)

Local Max / Min (in terms of y)Increasing /

Decreasing (in terms of x)

Approximate any local maxima or minima to the

nearest tenth.Find the intervals over which the

function is increasing and decreasing.

Find the Zeros of the polynomial function below

and sketch on the graph

Find the Zeros of the polynomial function below

and sketch on the graph

Multiplicity of 2 EVEN - Touches

Find the Zeros of the polynomial function below

and sketch on the graph

Find the Zeros of the polynomial function below

and sketch on the graph

NO X-INTERCEPTS!

Find the Zeros of the polynomial function below

and sketch on the graph