Warm-up5/8/08

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Warm-up 5/8/08

• Solve the system of linear equations.
• 4x 3y 12
• 2x 5y -20
• (0,4)

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Up Next

• Chapter on Quadratic Equations
• Getting ready for Algebra II
• Last test will be over this chapter.
• It will count as a regular test.
• For some of you, it will be your last chance to
  pull up your grade.

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Unit 7 Topic (Chapter 7)
Quadratic Equations and Functions
Key Learning(s)
How to set up and solve quadratic problems
Unit Essential Question (UEQ)
How do you solve quadratic equations and functions?
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Concept I
Modeling Data with Quadratic Functions
Lesson Essential Question (LEQ)
How do you graph and apply quadratic functions? How do you locate a quadratic function that is shifted?
Vocabulary
Parabola, quadratic functions, standard form, Axis of symmetry, vertex, maximum value
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Concept II
Square Roots
Lesson Essential Question (LEQ)
How do you use square roots when solving quadratics?
Vocabulary
Square root, principal, square root, negative square root, perfect squares
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Concept III
Solving Quadratic Equations
Lesson Essential Question (LEQ)
How do you determine whether a quadratic equation has two solutions, one solution, or no solutions?
Vocabulary

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Concept IV
Using the Quadratic Formula
Lesson Essential Question (LEQ)
When would you use the quadratic formula to solve a quadratic equation?
Vocabulary
Quadratic formula, vertical motion formula
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Concept V
Using the Discriminant
Lesson Essential Question (LEQ)
How do you use the discriminant to find the number of solutions of a quadratic equation?
Vocabulary
Discriminant
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Introduction

• On a piece of paper (in your notes)
• -Show the path of an arrow if it is aimed
  horizontally.
• -How does the path change if the arrow is aimed
  upward?
• -Name its shape

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7.1 Modeling Data withQuadratic Functions

• LEQ How can you tell before simplifying whether
  a function is linear, quadratic, or absolute
  value?
• Remember HOW?
• y (x 3)(x 2)
• f(x) x(x3)
• (x4)(x-7)
• (2b-1)(b-1)
• http//www.algebra.com/algebra/homework/Polynomial
  s-and-rational-expressions/Operations-with-Polynom
  ials-and-FOIL.lesson

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Quadratic Function

• A function that can be written in the form
• y ax2 bx c, where a?0.
• The graph of a quadratic function looks like a
  u (or part of one).
• The graph of a quadratic function is called a
• parabola.

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Linear or Quadratic?

• A function is linear if
• the greatest exponent of a variable is one
• A function is quadratic if
• the greatest exponent of a variable is two

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Linear or Quadratic?

1. y (x 3)(x 2)
2. f(x) x(x 3)
3. f(x) (x2 5x) x2
4. y (x 5)2
5. y 3(x 1)2 4
6. h(x) (3x)(2x)
7. f(x) ½ (4x 10)
8. y 2x (3x 5)
9. f(x) -x(x 4) x2
10. y -7x

• Quadratic
• Quadratic
• Linear
• Quadratic
• Quadratic
• Quadratic
• Linear
• Linear
• Linear
• Linear

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Ex1)

• Make a table of values and graph the quadratic
  functions y 2x2 and y -2x2.
• What is the axis of symmetry for each graph?
• What effect does a negative sign have on the
  shape of a quadratic functions graph?
• Graph y -1/3 x2, y ½ x2, y -x2
• Compare the width of the graphs above.
• How could you quickly sketch the graph of a
  quadratic equation?

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Properties of parabolas

• Important parts
• Vertex
• The point at which the function has a maximum or
  minimum
• Axis of Symmetry
• Divides a parabola into two parts that are mirror
  images

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Maximum or Minimum

• If a parabola opens down, does it have a maximum
  or minimum for the vertex?
• If a parabola opens up, does it have a maximum or
  a minimum for the vertex?

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Practice

• Section 7.1
• p. 321-322
• 2 40 Even
• Assignment
• Section 7.1
• p. 321 322
• 1 11 odd, 17, 19, 25, 29, 31, 35 - 39

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Warm-up 2/19/08

• Re-write each equation without parenthesis.
• Y (x 1)2 4
• Y 2(x 5)2
• Y -(x 3)2 6
• Y -3(x 7)2
• Y 7(x 4)2

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Reminders

• Late projects?
• Late word problems?

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Assignment

• Long weekend homework?
• Section 5.2
• p. 208-209
• 1-31 odd

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Refresh

• Graphic organizer (vertex form)

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Warm-up 2/20/08Write the equation of the
parabola shown (in vertex and standard form)
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Write the equation of the parabola shown
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Homework

• p. 210
• 38 40, 42
• Worksheet
• Practice

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5.3 Vertex vs. Standard Form

• LEQ1 How do you find the vertex form of a
  function written in standard form?
• LEQ2 How do you transfer functions from vertex
  to standard form?
• Which is better, standard or vertex form?
• Standard Form Vertex Form
• Y -3x2 12x 8 y -3(x 2)2 4

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• When the equation of a function is written in
  standard form,
• 1) the x-coordinate of the vertex is b/2a.
• 2) To find the y-coordinate, you substitute the
  value of the x-coordinate for x in the equation
  and simplify.

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Ex1.

• Write the function y 2x2 10x 7 in vertex
  form.
• x-coordinate -b/2a -10/2(2) -10/4
• -5/2 or - 2.5
• y-coordinate 2(-2.5)2 10(-2.5) 7
• -5.5
• Substitute the vertex point (-2.5,-5.5) into the
  vertex form y a(x h)2 k a from above
• y 2(x 2.5)2 5.5

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Ex2.

• Write y -3x2 12x 5 in vertex form.
• x-intercept -b/2a -12/2(-3) -12/-6 2
• y-intercept y -3(2)2 12(2) 5 17
• Re-write in vertex form
• y -3(x 2)2 17

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Write in vertex form
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Write in vertex form
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Write in vertex form
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Write in vertex form
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From vertex to standard form

• To change an equation from vertex to standard
  form, you have to multiply out the function.
• y 3(x -1)2 12
• y 3(x 1)(x 1) 12
• y 3(x2 2x 1) 12
• y 3x2 6x 3 12
• y 3x2 6x 15

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• Graphic Organizer

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Assignment

• Section 5.3
• p.212-213
• 1-4 all, 8-18 even,
• 22-30 even, 36

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Warm-up 5.3

• The Smithsonian Institution has a traveling
  exhibit of popular items. The exhibit requires 3
  million cubic feet of space, including 100,000
  square feet of floor space. How tall must the
  ceilings be?
• The exhibit also requires a constant temperature
  of 70ºF, plus or minus 3º. write an inequality
  to model this temperature, T.

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Warm-up 5.4 5.5

• Find the inverse of each equation.
• y -v(x 2)
• y - vx -1
• y v(x2) 6
• y v(x 3) 2
• y v(x 1) - 4

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Warm-up 5.5

• Multiply the two binomials
• (y 7 )(y 3)
• (2x 4)(x 9)
• (4b 3)(3b 4)
• (6g 2) (g 9)
• (7k 5)(-4k 3)

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5.5 Factoring Quadratic Equations

• LEQ When is the quadratic formula a good method
  for solving an equation?
• Which is better, standard or vertex form?
• One way to solve a quadratic equation is to
  factor and use the Zero-Product Property.
• For all real numbers a and b, if ab 0 then a
  0 or b 0.

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• To solve by factoring, first write an equation
  in standard form.
• Factoring x2 5x 6 requires you find two
  binomials of the form (x m)(x n), whose
  product is x2 5x 6.
• M and N must have a sum of 5 and a product of 6.

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• Steps
• List the factor pairs whose product is 6
• Find two of those factors whose sum is 5
• Ex. Factor pairs of 6
• 6 x 1 2 x 3 -1 x -6 -2 x -3
• Only one pair has a sum of 5 2 and 3
• Thus, m n are 2 3
• (x 2)(x 3)
• Always check by using the FOIL method!

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• Ex. Factor x2 7x 12
• Find factor pairs of 12
• 1x12 2x6 3x4 -1x-12 -2x-6 -3x-4
• Which factor pair has a sum of -7?
• -3x-4
• So, put -3 and -4 in where the m n would be.
• (x 3)(x 4)
• Check with FOIL.

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Factor the trinomials.

1. x2 12x 20
2. x2 9x 20

• (x 10)(x 2)
• (x 5)(x 4)

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Warm-up 5.5cont.Find the inverse of each
function.

1. v(x/2 1)
2. Y -1/2 x
3. Y 1/3x2 - 2

1. Y 2x2 2
2. Y -2x
3. Y v(3x 6)

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5.5 Continued Solve each equation by factoring.

1. x2 6x 8 0
2. x2 2x 3
3. 2x2 6x -4

• X -4,-2
• X 3, -1
• X -1,-2

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Solving by finding roots

• Quadratic equations can also be solved by
  finding the square roots.
• This method is effective when theres no b.
• Ex. 0 -16x2 1600
• -1600-16x2
• 16 16
• 100 x2
• x 10
• Determine the reasonableness of a negative
  answer based on the situation.

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Word Problem

• A smoke jumper jumps from a plane that is 1700
  ft above the ground. The function
• y -16x2 1700 gives a jumpers height y in
  feet after x seconds.
• How long is the jumper in free fall if the
  parachute opens at 1000 ft?
• How long is the jumper in free fall if the
  parachute opens at 940 ft?

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Roots using the calculator

• Roots, also called zeros are really the points
  where a quadratic equation intercepts the x-axis
  (where x 0).
• To find the zeros using a calculator
• 1) enter the quadratic function under y
• 2) 2nd calc zeros
• 3) left bound? Right bound? Enter

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Find the roots of each equation by graphing.
Round answers to tenths

1. x2 7x -12
2. 6x2 -19x 15
3. 5x2 7x 3 8
4. 1 4x2 3x

1. X 3,4
2. X -1.5, -1,7
3. X -0.9, 2.3
4. X -1, 0.3

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What if you cant factor it?

• If youre having trouble factoring a quadratic
  equation, you can always use the quadratic
  formula.
• (Graphic Organizer)

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Solve using any method.

1. 5x2 80
2. X2 11x 24 0
3. 12x2 154 0
4. 2x2 5x 3 0
5. 6x2 13x 6 0
6. X2 8x - 7

1. X 4, -4
2. X 3, 8
3. X 3.6, - 3.6
4. X 3, - 0.5
5. X -1.5, -0.7
6. X 7, 1

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T.O.T.D.

• Answer the LEQs.
• 1) When is the quadratic formula a good method
  for solving an equation?
• 2) Which form is better? Standard or vertex
  form? Are some situations easier to use one or
  the other? Explain.
• (this question comes from several days)

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Warm-up 5.6

• Solve each equation. Give an exact answer if
  possible. Otherwise write the answer to two
  decimal places.
• x2 4x 21 0
• 2x2 3x 0
• (x 3)(x 4) 12
• (x 1)(x 2)(2x 1) 0

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What if you cant factor it?

• If youre having trouble factoring a quadratic
  equation, you can always use the quadratic
  formula.
• (Graphic Organizer)

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5.6 Complex Numbers

• LEQ How are complex numbers used in solving
  quadratic equations?
• What do you know about the graph?
• From previously, what if you had the equation
• x2 25 0
• You end up taking the v of a negative number!
  (Calculator wont work)

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The Imaginary Number

• In order to deal with the negative square root,
  the imaginary number was invented.
• Imaginary Number i defined as v-1
• For now, youll probably only use imaginary
  numbers in the context of solving quadratics for
  their zeros.
• From the web

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Imaginary Number

• i is the symbol for the imaginary number.
• It is a complex number whose square root is
  negative or zero.
• Rene Descartes was coined the term in 1637 in his
  book La Giometrie.
• The numbers are called imaginary because they are
  not always applied in the real world.

i
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Imaginary Number Applications

• In electrical engineering, when looking at AC
  circuitry, the values of electrical voltage are
  expressed as complex imaginary numbers known as
  phasors.
• Imaginary numbers are used in areas such as
  signal processing, control theory,
  electromagnetism, quantum mechanics and
  cartography.

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Imaginary Number

• In mathematics Imaginary Numbers,also called an
  Imaginary Unit, can be found when working with
  quadratic functions.
• An equation like x210 has an imaginary root,
  and requires the use of the quadratic formula to
  solve it.

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The Discriminant

• Whether or not you end up with a complex number
  as an answer depends solely on the discriminant.
• The discriminant refers to the part of the
  quadratic equation that is under the square root.

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Nature of the solutions

• I. If the discriminant is positive
• -There are two real solutions
• -The graph of the equation crosses the x-axis
  twice (has two zeros)
• If the discriminant is zero
• -There is one real solution
• -the graph of the equation only touches the
  x-axis once (has one zero)

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• If the discriminant is negative
• -There is no real solution
• -There are two imaginary solutions
• -The graph never touches the x-axis.
• Example 1 y x² 2x 1
• a 1 b 2 c 1
• Discriminant 2² - 4 1 1 4 - 4 0
• Since the discriminant is zero, there should be 1
  real solution to this equation.
• Also, the graph only touches the x-axis once.

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Calculate the discriminant to determine the
number and nature of the solutions of the
following quadratic equation

1. x² - 2x 1
2. y x² - x - 2
3. y x² - 1
4. y x² 4x - 5
5. y x² 4x 5
6. y x² 4
7. y x² 25

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Simplifying complex numbers

1. i2 (v-1)(v-1) -1
2. i3
3. i4
4. i5
5. i6
6. i7

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Complex Numbers

• A number of the form a b(i) , where a and b
  are real numbers, is called a complex number.
  Here are some examples
• 2 i, 2 v3i
• The number a is called the real part of abi, the
  number b is called the imaginary part of abi.

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Operations with Complex Numbers

• Adding Subtracting them Just like combining
  like terms
• Ex. 3i -1i 2i
• (5 7i) (-2 6i) Combine like terms,
  simplify
• 5 2 7i 6i
• 3 13i

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Multiplying

• Distribute, combine like terms, simplify
• Ex) (5 7i)(-2 6i)
• 10 30i 14i 42i2
• 10 16i 42(-1)
• 10 16i 42
• -32 16i

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Warm-up 5.7

• Simplify each expression
• 1) v-25
• (2 3i)(3 4i)
• Multiply.
• (x 1)(x 1) 5) (x 6)(x 6)
• (x 3)(x 3) 6) (2x 1)(2x 1)

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5.7 Completing the square

• LEQ How is completing the square useful when
  solving quadratic equations?
• Binomial Squared
• (x 5)2 x2 2(5)x 52 x2 10x 25
• (x 4)2 x2 2(-4)x (-4)2 x2 8x 16
• (x b/2)2 x2 2(b/2)x (b/2)2
• x2 bx (b/2)2

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• The process of finding the last term of a
  perfect square trinomial is called completing the
  square.
• This method is useful for making vertex form.
• Find the b-term.
• Divide the b-term by 2
• The square of this will be c.
• Try these
• 1) x2 2x ___ 2) x2 12x ___

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Ex1.

• Solve by completing the square.
• x2 8x 36
• Write the equation with all x-terms on one side
• x2 8x - 36
• Complete the square (add to both sides)
• x2 8x (-4) 2 -36 (-4)2
• Re-write (x 4)2 -36 16
• (x 4)2 -20
• It would be easy to graph this in vertex form.

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• To solve the equation, continue algebraically.
• (x 4)2 -20
• v(x 4)2 v-20
• (x 4) v-20
• x 4 v-20
• x 4 v(2)(2)(5)I
• x 4 2iv5
• The two solutions are
• x 4 2iv5
• x 4 2iv5

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Assignment

• Quiz
• 5.7

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Warm-up Test

1. Classify 3x(2x) as linear, constant, or
   quadratic.
2. Re-write the equation of the parabola in vertex
   form y x2 8x 12.
3. Find the absolute value of 6 9i.
4. Simplify -2(2 4i) 8(5 2i).
5. Find the coordinates of the vertex for the graph
   of y x2 10x 2

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Warm-up Test (2)

1. How do you use the vertical line test to
   determine if a graph represents a function?
2. Find f(g(2)) if f(x) 2x 2 and g(x) 9x.
3. Graph the inequality y gt 2x 1.
4. How do you solve a system of three equations in
   three variables?
5. http//apps.collegeboard.com/qotd/question.do