Advanced Algebra: Chapter 5

1
Advanced Algebra Chapter 5

• Quadratic Functions

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Graphing Quadratics5.1
3
Quadratic Functions

• Quadratics are in the form
• The graph is called a parabola
• U-shaped
• Vertex
• The lowest or highest point of the graph
• Axis of Symmetry
• Vertical line thru the vertext

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Graphing

• The graph of is a
  parabola with the following characteristics
• If a is positive
• Opens up
• If a is negative
• Opens Down
• The x-coordinate of the vertex
• The axis of symmetry is the vertical line

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Graphing

• Example

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Graphing

• Example

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Graphing

• Example

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Vertex and Intercept Forms

• Vertex Form
• Vertex is (h , k)
• Axis of Symmetry is x h

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Vertex and Intercept Forms

• Intercept Form
• x-intercepts are p and q
• The axis of symmetry is halfway between (p ,
  0) and (q , 0)

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Graphing

• Example

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Graphing

• Example

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Writing Equations in Standard Form

• Distribute!

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Writing Equations in Standard Form

• Distribute!

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p.253 21-31 Odd, 38-43
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Solving By Factoring5.2
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Factoring

• Binomials
• Trinomial
• Factoring
• Write a trinomial as a product of binomials

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Factoring .

• First, list all of the factors of c
• The factors of c that add to b are your solutions

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Examples
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Examples
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Examples
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Factoring .

• Factors into where k
  and l are factors of a and m and n are factors
  of c
• Trial and error

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Examples
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Examples
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Examples
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p.261 29-40
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Solving QuadraticsDay 2
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Factoring with Special Patterns

• Difference of Two Squares
• Perfect Square Trinomial

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Examples
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Examples
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Examples
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Examples
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Examples
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p.26148-72 Even
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Solving EquationsDay 3
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Zero Product Property

• If AB 0, then A 0 or B 0, or both
• The key is for the equation to equal zero!
• Examples

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Examples
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Examples
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Examples
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Examples
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Examples
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p.262 74-88
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Solving Quadratics by Finding Square Roots5.3
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Square Roots

• A number r is a square root of a number if
• Any positive number has two square roots
• Positive
• Negative
• Example

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Properties of Square Roots

• Product Property
• Quotient Property

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Properties of Square Roots

• Does addition or subtraction work???
• Absolutely Not!
• Ex

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Examples

47
Simplifying Radicals

• Rationalizing the Denominator
• Eliminating the radical from the bottom of a
  fraction
• Standard form of a radical expression

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Examples


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Solving Quadratics

• First, isolate the squared variable
• i.e. solve for the variable
• Find the square root of both sides

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Examples
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Examples
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Examples
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Examples
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p.267 20-58 Even
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Complex Numbers5.4
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Complex Numbers

• A complex number is an expression of the
  formwhere a and b are real numbers and The
  real part of the complex number is a and the
  imaginary part is b.

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What is i?

• Imaginary numbers

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Complex Numbers

• Two complex numbers are equal iff their real
  parts are equal and their imaginary parts are
  equal

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Graphing Complex numbers

• 3 2i
• 2 2i
• -3 2i
• - i

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Complex Numbers

• Addition
• Subtraction
• Multiplication

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Complex Numbers

• Division

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Examples

• (3 5i) (4 2i)
• (3 5i)(4 2i)

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Examples

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Examples
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Square Roots of Neg. s

• If r is negative, then the principal square root
  of r isThe two square roots of r are
  and

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Examples
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Examples
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Examples
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Examples
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p.277 17-55 Odd
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Complex Numbers and Conjugates
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What is a conjugate?

• A conjugate of a complex number is a number with
  the same real part but has opposite sign of the
  imaginary part
• Ex
• 3 4i 3 4i
• 2 7i 2 7i

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Complex Numbers

• Division of Complex Numbers

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Example
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Example
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Example
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Absolute Value

• The absolute value of a complex numberis a
  nonnegative real number defined as
• Absolute Value is the distance from zero
• Origin

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Example

• Find the absolute value of the complex number 3
  2i

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Example

• Find the absolute value of the following

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Example

• Find the absolute value of the following

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p.278 56-71
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Completing the Square5.5
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Completing the Square

• Allows you to write an expression in the form
  as the square of a binomial

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Completing the Square
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Completing the Square
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Completing the Square
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Completing the Square
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Completing the Square
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Completing the Square
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Completing the Square
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Completing the Square

• Sohow does this work algebraically?
• This only works when the leading coefficient is 1

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Completing the Square

• Solve for c

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Completing the Square

• Solve for c

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Completing the Square

• Solve for c

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Completing the Square

• Make the leading coefficient 1
• Move the c value to the other side
• Decide what is needed to complete the square and
  make it happen
• Solve

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Completing the Square
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Completing the Square
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Completing the Square
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Completing the Square
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p.286 32-37, 47-51, 55-59
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The Quadratic Formula5.6
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The Quadratic Formula

• Can use this to factor ANY quadratic expression.
• Will work for both real and imaginary zeros

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The Quadratic Formula
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The Quadratic Formula

• Factor the following

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The Quadratic Formula

• Factor the following

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The Quadratic Formula

• Factor the following

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The Quadratic Formula

• Factor the following

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The Quadratic Formula

• Factor the following

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The Quadratic Formula

• Factor the following

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The Quadratic Equation

• Is there a pattern occurring from these examples?
• What did the equations with 2 real values have in
  common?
• With 1 real value?
• With no real values?

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The Discriminant

• The equation has 2 real
  solutions
• The equation has 1 real
  sol.
• The equation has 2
  imaginary sol.

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p.295 32-50
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Pop Quiz! ?
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Graphing and Solving Quadratic Inequalities5.7
115
Steps for Graphing Inequalities

• Draw the parabola as a regular equation
• Solid for equal to open for not equal to
• Chose a point inside the parabola
• If true Shade inside
• If false Shade outside

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Example
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Example
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Systems of Inequalities

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Systems of Inequalities

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Finding Solutions of One Variable

• When given an equation we try to find values of
  x that make the equation equal zero
• X-intercepts
• Roots
• Same with quadratics, however now finding a
  range(s) that make the inequality true

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Examples
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Examples
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p.30323-31, 41,42
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Modeling Quadratics5.8
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Writing Quadratics From Graphs

• Choosing how to write a graph depends on the info
  we have
• If we have the vertex and a point
• Vertex form
• If we have the intercepts and another point
• Intercept form

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Vertex Form
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Intercept Form
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Finding Quadratic Models

• A study comparing speed and mpg shows
• Find the graph representingthe data

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p.309 8-26 Even