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Splash Screen
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6.1 Graphing Quadratic Functions

• Quadratic function equation in the following
  form
• ax2 bx c 0 a ? 0
• Graph is called a parabola

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Example 1-1a
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Example 1-1b
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Parts of a quadratic function

• Axis of symmetry
• A line where the parabola can be folded and be
  the exact same thing on both sides
• Equation

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• Vertex
• The point where the axis of symmetry meets the
  parabola
• X coordinate
• Y intercept
• Where the parabola crosses the y axis
• The value of c

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Example 1-2a
Find the y-intercept, the equation of the axis of
symmetry, and the x-coordinate of the vertex.
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Example 1-2b
Consider the quadratic function a. Find the
y-intercept, the equation of the axis of
symmetry, and the x-coordinate of the
vertex. b. Make a table of values that includes
the vertex. c. Use the information to graph the
function.
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Maximum/Minimum

• The y coordinate of a quadratic function is the
  maximum value or minimum value
• If a gt 0 (positive), then the parabola opens up
  and has a minimum value
• If a lt 0 (negative), then the parabola opens down
  and has a maximum value

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Example 1-3a
Determine whether the function has a maximum or a
minimum value.
State the maximum or minimum value of the
function.
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Example 1-3b
Answer minimum
Answer 5
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6.2 Solving Quadratic Equations by Graphing

• The solutions of a quadratic equation are called
  roots of the equation
• One method for finding the roots of a quadratic
  equation is to find the zeros of the function
  meaning where y 0

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Example 2-1a
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Example 2-1b
Answer
3 and 1
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Number of Solutions

• One Real Solution
• Two Real Solutions
• No Real Solutions

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Example 2-2a
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Example 2-2b
Answer 3
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Example 2-3a
Number Theory Find two real numbers whose sum
is 4 and whose product is 5 or show that no such
numbers exist.
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Example 2-3b
Number Theory Find two real numbers whose sum
is 7 and whose product is 14 or show that no
such numbers exist.
Answer no such numbers exist
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Estimate Solutions

• Sometimes exact solutions (whole numbers) can not
  be found
• Here we say what two numbers the solution is
  between

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Example 2-4a
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Example 2-4b
Answer between 0 and 1 and between 3 and 4
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Classwork
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3/24/2009 Warm - Up
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Homework Review

• Section 6.1
• 15 x 4 4
• 12 x -2 -2
• 1 x 0.5 0.5
• Min -9
• Min 5
• Max -8
• Min -8
• Max 3
• Max 0

• Section 6.2
• -1 1
• No real solutions
• 1 2
• b/t 0 1 b/t -4 -3
• -6 -4
• b/t -2 -1 2

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Homework Review
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6.3 Solving Quadratic Equations by Factoring

• Remember
• You have a quiz tomorrow on 6.1 6.3
• You will have an assignment at the end of the
  lesson that will be practice for this quiz
• It will be taken up for a grade at the end of
  class ?

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Zero Product Property

• Another way to solve quadratic equations is by
  factoring.
• Zero Product Property For any numbers, if a b
  0, then one or both numbers have to be 0
• We will factor quadratic equations, then set all
  factors 0 and solve them. (Always check!)

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Example 3-1a
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Example 3-1a
Solve x2 6x by factoring.
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Example 3-1a
Solve x2 6x 16 0 by factoring.
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Example 3-1a
Solve x2 9 6x by factoring.
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Example 3-1a
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Example 3-1a
Solve 2x2 7x 15 by factoring.
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Example 3-1b
Solve each equation by factoring. a. b. c.
x2 4x 21
Answer 0, 3
Answer -3, 7
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Double Roots

• When the quadratic equation has only one solution
  (graph only touches the x axis once or kisses
  it) then we have what is called a double root
  (because it occurs twice)
• When we factor and our factors are the same, this
  means there is a double root, and only one real
  solution
• We can always check ourselves by looking at the
  graph of the quadratic equation

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Example 3-2a
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Example 3-2b
Answer 5
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Example 3-3a
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Example 3-3b
Answer C
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Writing quadratic equations

• You may be given solutions and asked to write the
  quadratic equation
• You do this by working backwards
• Set x to the solution(s)
• Move the solution(s) to where the equation 0
• Multiply (FOIL or BOX) the equation(s)
• Simplify as needed

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Example 3-4a
Write a quadratic equation with and 6 as
its roots. Write the equation in quadratic form.
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Example 3-4b
Write a quadratic equation with and 5 as
its roots. Write the equation in quadratic form.
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Classwork

• The following assignment needs to be completed by
  the end of class as a review for your quiz
  tomorrow.
• Once you have finished this you are done for the
  day
• Homework
• Study 6.1 6.3
• Quiz first thing tomorrow ?

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3/25/2009 Warm - Up
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Quiz Review
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Quiz Time

• You will have 30 35 minutes to complete this
  quiz
• Take your time
• Good Luck!
• We will begin 6.5 once everyone has finished we
  will do 6.4 tomorrow ?

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6.5 The Quadratic Formula and the Discriminant

• Just like graphing and factoring, the quadratic
  formula can give us the solution to a quadratic
  equation (proof pg. 313)
• Quadratic Formula
• Just like before you can check yourself by
  looking at the graph

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Example 5-1a
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Example 5-1a
Solve x2 12x 28 by using the Quadratic
Formula.
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Example 5-1b
Answer 2, 15
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Example 5-2a
Identify a, b, and c. Then, substitute these
values into the Quadratic Formula.
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Example 5-2b
Solve by using the Quadratic Formula.
Answer 11
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Example 5-3a
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Example 5-3b
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Example 5-4a
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Example 5-4b
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Roots and the Discriminant

• Discriminant b2 4ac
• What is under the radical
• The discriminant determines
• How many roots you have
• One or Two
• What type of roots you have
• Real, Rational, Irrational, Complex

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Value of Discriminant Type/Number of Roots
Discriminant is positive and a perfect square 2 real, rational roots
Discriminant is positive and not a perfect square 2 real, irrational roots
Discriminant is zero 1 real, rational root
Discriminant is negative 2 complex roots
Example of a Graph



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Example 5-5a
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Example 5-5a
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Example 5-5a
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Example 5-5a
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Example 5-5b
Answer 0 1 rational root
Answer 24 2 complex roots
Answer 5 2 irrational roots
Answer 64 2 rational roots
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Which method do I use?

• If the problem does not tell me which method to
  use, how do I know which on is the easiest,
  quickest, and most efficient method to use?

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Solving Quadratic Equations (pg 317)
Method Can be Used When to Use
Graphing Sometimes If an exact answer is not required best to use to check yourself
Factoring Sometimes If c 0 or factors are easy to see
Square Root Property Sometimes If an equation that is a perfect square is equal to a constant
Completing the Square Always If b is an even number
Quadratic Formula Always If all other methods fail or are too tedious
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Classwork/Homework
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Example 4-1a
Answer The solution set is 15, 1.
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Example 4-1b
Answer 3, 13
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Example 4-2a
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Example 4-2a
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Example 4-2a
Check Use the ZERO function of a graphing
calculator. The approximate zeros of the related
function are 1.5 and 8.5.
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Example 4-2b
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Example 4-3a
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Example 4-3b
Answer 9 (x 3)2
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Example 4-4a
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Example 4-4a
Answer The solution set is 6, 2.
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Example 4-4b
Answer 6, 1
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Example 4-5a
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Example 4-5a
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Example 4-5a
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Example 4-5b
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Example 4-6a
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Example 4-6a
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Example 4-6a
Check A graph of the related function shows that
the equation has no real solutions since the
graph has no x-intercepts. Imaginary solutions
must be checked algebraically by substituting
them in the original equation.
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Example 4-6b
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End of Lesson 4
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Lesson 6 Contents
Example 1 Graph a Quadratic Function in Vertex
Form Example 2 Write y x2 bx c in Vertex
Form Example 3 Write y ax2 bx c in Vertex
Form, a ? 1 Example 4 Write an Equation Given
Points
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Example 6-1a
h 3 and k 2
Now use this information to draw the graph.
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Example 6-1a
Step 1 Plot the vertex, (3, 2).
(1, 6)
(5, 6)
(2, 3)
(4, 3)
Step 3Find and plot two points on one side of
the axis of symmetry, such as (2, 3) and (1,
6).
(3, 2)
Step 4Use symmetry to complete the graph.
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Example 6-1b
Answer
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Example 6-2a
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Example 6-2a
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Example 6-2b
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Example 6-3a
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Example 6-3a
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Example 6-3a
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Example 6-3b
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Example 6-4a
Write an equation for the parabola whose vertex
is at (1, 2) and passes through (3, 4).
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Example 6-4a
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Example 6-4a
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Example 6-4b
Write an equation for the parabola whose vertex
is at (2, 3) and passes through (2, 1).
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End of Lesson 6
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Lesson 7 Contents
Example 1 Graph a Quadratic Inequality Example
2 Solve ax2 bx c ? 0 Example 3 Solve ax2 bx
c ? 0 Example 4 Write an Inequality Example
5 Solve a Quadratic Inequality
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Example 7-1a
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Example 7-1a
Step 2Test a point inside the parabola, such as
(1, 2).
(1, 2)
So, (1, 2) is a solution of the inequality.
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Example 7-1a
Step 3 Shade the region inside the parabola.
(1, 2)
(1, 2)
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Example 7-1b
Answer
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Example 7-2a
The solution consists of the x values for which
the graph of the related quadratic function lies
above the x-axis. Begin by finding the roots of
the related equation.
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Example 7-2a
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Example 7-2b
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Example 7-3a
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Example 7-3a
Sketch the graph of the parabola that has
x-intercepts of 3.16 and 0.16. The graph should
open down since a lt 0.
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Example 7-3a
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Example 7-3a
Check Test one value of x less than 3.16, one
between 3.16 and 0.16, and one greater than 0.16
in the original inequality.
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Example 7-3b
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Example 7-4a
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Example 7-4a
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Example 7-4a
Answer Thus, the ball is within 15 feet of the
ground for the first 0.46 second of its flight
and again after 2.04 seconds until the ball hits
the ground at 2.5 seconds.
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Example 7-4b
Answer The ball is within 10 feet of the ground
for the first 0.5 second of its flight and again
after 1.25 seconds until the ball hits the ground.
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Example 7-5a
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Example 7-5a
Plot 2 and 1 on a number line. Use closed
circles since these solutions are included.
Notice that the number line is separated into 3
intervals.
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Example 7-5a
Test a value in each interval to see if it
satisfies the original inequality.
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Example 7-5a
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Example 7-5b
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End of Lesson 7
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