Quadratic Equations and Functions

1
Chapter 5

• Quadratic Equations and Functions

2
5-1 Warm Up

• What is a quadratic equation? What does the graph
  look like? Give a real world example of where it
  is applied.

3
5-1 Modeling Data with Quadratic Functions

• OBJ Recognize and use quadratic functions
• Decide whether to use a linear or a quadratic
  model

4
Quadratic Functions

• Quadratic function is a function that can be
  written in the form f(x) ax²bxc, where a ? 0
• The graph is a parabola
• The ax² is the quadratic term
• The bx is the linear term
• The c is the constant term

5

• The highest power in a quadratic function is two
• A function is linear if the greatest power is one

6
Tell whether each function is linear or quadratic

• F(x) (-x3)(x-2)
• Y(2x3)(x-4)
• F(x)(x²5x)-x²
• Y x(x3)

7
Modeling Data

• Last semester you modeled data that, when looking
  at the scatter plot, the data seemed to be linear
• Some data can be modeled better with a quadratic
  function

8
Find a quadratic model that fits the weekly sales
for the Flubbo Toy Comp
9
5-1 Wrap Up

• What is a quadratic function?
• What kinds of situations can a quadratic function
  model?

10
5-2 Warm Up

• List as many things as you can that have the
  shape of a parabola

11
5-2 Properties of Parabolas

• OBJ Find the min and max value of a quadratic
  function
• Graph a parabola in vertex form

12
Comparing Parabolas

• Any object that is tossed or thrown will follow
  a parabolic path.
• The highest or lowest point in a parabola is the
  vertex
• It is the vertex that is the maximum or minimum
  value
• If a is positive the parabola opens up, making
  the vertex a min point
• If a is negative the parabola opens down, making
  the vertex a max point

13

• Axis of symmetry divides a parabola into two
  parts that are mirror images of each other
• The equation of the axis of symmetry is x(what
  ever the x coordinate of the vertex is)
• Two corresponding points are the same distance
  from the axis of symmetry

14

• Yax²bxc is the general equation of a parabola
• If agt0 the parabola opens up
• If alt0 the parabola opens down
• If a is a fraction it is a wide opening
• If a is a whole number it is narrow

15
Examples

• Each tower of the Verrazano Narrows Bridge rises
  about 650 ft above the center of the roadbed. The
  length of the main span is 4260 ft. Find the
  equation of the parabola that could model its
  main cables. Assume that the vertex of the
  parabola is at the origin.

16
Translating Parabolas

• Not every parabola has its vertex at the origin
• Ya(x-h)²k is the vertex form of a parabola
• It is a translation of yax²
• (h,k) are the coordinates of the vertex

17
(No Transcript)
18
Sketch the following graphs
19

• The vertex is ( h, k)
• Axis of symmetry is x h
• If a gt 0 it is a max
• If a lt 0 it is a min

20
Graph, give equation for axis of symmetry and
state the vertex.
21
Example

• Sketch the graph of y -1/2(x-2)²3
• Sketch the graph of y 3(x1)²-4

22
Wrap Up 5-2

• What does the vertex form of a quadratic function
  tell you about its graph?

23
Warm Up 5-3

• List formulas that you know to use to find
  answers to problems quickly. (list as many
  formulas as you can)

24
5-3 Comparing Vertex and Standard Forms

• OBJ Find the vertex of a function written in
  standard form
• Write equations in vertex and standard form

25
Get into a group of four

• Turn to page 211
• Do part a
• What do you notice about the graphs of each pair
  of equations?
• What is true of each pair of equations?
• Write a formula for the relationship between b
  and h
• How can we modify our formula to show the
  relationship among a, b, and h. (the last couple
  of equations)

26
Standard form of a parabola

• When a parabola is written as yax²bxc it is
  standard form
• The x coordinate of the vertex can be found by
  b/(2a)
• To find the y coordinate by (b2-4ac)/4a

27

• Suppose a toy rocket is launched to its height in
  meters after t seconds is given by H -4.9t2
  20t 1.5. How high is the rocket after one
  second? How high is the rocket when launched. How
  high is the rocket after 12 seconds?

28
Example

• Write the function y 2x²10x7 in vertex form
• Write the function y -x²3x-4 in vertex form
• What is the relationship between the axis of
  symmetry and the vertex of the parabola?

29
Example

• As a graduation gift for a friend, you plan to
  frame a collage of pictures. You have a 9 ft
  strip of wood for the frame. What dimensions of
  the frame give you maximum area of the collage?
• What is the maximum area for the collage?
• What is the best name for the geometric shape
  that gives the maximum area for the frame?
• Will this shape always give the max area?

30
Consider this general formula
31

• A ball is dropped form the top of a 20 meter tall
  building. Find an equation describing the
  relation between the height and time. Graph its
  height h after t seconds. Estimate how much time
  it takes the ball to fall to the ground. Explain
  your reasoning

32

• Write y 3(x-1)²12 in standard form
• A rancher is constructing a cattle pen by a
  river. She has a total of 150 ft of fence, and
  plan to build the pen in the shape of a
  rectangle. Since the river is very deep, she
  need only fence three sides of the pen. Find the
  dimensions of the pens so that it encloses the
  max area.

33
(No Transcript)
34

• Suppose a swimming pool 50 m by 20 m is to be
  built with a walkway around it. IF the walkway is
  w meters wide, write the total area of the pool
  and walkway in standard form

35
Consider this

• If a quarterback tosses a football to a receiver
  40 yards downfield, then the ball reaches a
  maximum height halfway between the passer and the
  receiver, it will have a equation

36
Example

• Suppose a defender is 3 yards in front of the
  receiver. This means the defender is 37 yards
  from the quarterback. Will he be able deflect or
  catch the ball?

37
Examples

• A model rocket is shot at an angle into the air
  from the launch pad. The height of the rocket
  when it has traveled horizontally x feet from the
  launch pad is given by

38

• A 75-foot tree, 10 feet from the launch pad is in
  the path of the rocket. Will the rocket clear the
  top of the tree?
• Estimate the maximum height the that the rocket
  will reach.

39
Wrap Up 5-3

• Describe the similarities and differences between
  the vertex form and standard form of quadric
  equations.

40
Warm Up 5-4

• Name mathematical operations that are opposites
  of each other. For example, addition is the
  opposite of subtraction.
• Two inverse functions are opposite of each other
  in the same way.

41
5-4 Inverses and Square Root Functions

• OBJ Find the inverse of a function
• Use square root functions

42
Consider the functions

• F(x) 2x-8
• G(x) (x8)/2
• Find F(6) and G(4)
• F(x) and G(x) are inverses because one function
  undoes the other
• Graph each function on the same coordinate plane
• Find three coordinates on f(x)
• Reverse the coordinates and graph
• What do you notice?

43
Definition

• The inverse of a relations is the relation
  obtained by reversing the order of the
  coordinates of each ordered pair in the relation

44
Remember

• If the graph of a function contains a point
  (a,b), then the inverse of a function contains
  the point (b,a)

45
Example
46
Inverse Relation Theorem

• Suppose f is a relation and g is the inverse of
  f. Then
• A rule for g can be found by switching x and y
• The graph of g is the reflection image of the
  graph of f over yx
• The domain of g is the range of f, and the range
  of g is the domain of f

47
Remember

• The inverse of a relation is always a relation
• The inverse of a function is not always a function

48
Examples

• Consider the function with equation y 4x-1. Find
  an equation for its inverse. Graph the function
  and its inverse on the same coordinate plane. Is
  the inverse a function?

49

• Consider the function with domain the set of all
  real numbers and equation yx2
• What is the equation for the inverse? Graph the
  function and its inverse on the same coordinate
  plane. Is the inverse a function? Why or Why not?

50
Example

• Graph the function and its inverse. The write the
  equation of the inverse

51
More Examples

• Find the inverses of these functions

52
Square Root Functions

• Y?x is the square root function
• The graph starts at (0,0)
• The domain is x?0
• The range is y?0

53
Example

• Graph the function and state the domain and range

54
5-4 Wrap Up

• What can you tell me about a function and its
  inverse?

55
Warm Up 5-5

• Brain storm all the methods you know for solving
  this equation. Include less efficient methods. We
  will vote on which you all prefer.

56
5-5 Quadratic Equations

• OBJ Solve quadratic equations by factoring,
  finding square roots, and graphing

57
Zero Product Property

• For all real numbers a and b. If ab0, then a0
  or b0
• Example
• (x3)(x-7)0
• (x3)0 or (x-7)0

58

• We can use the Zero Product Property to solve
  quadratic equations.
• That is why we learned how to factor.

59
Important rule for factor quadratics

• Make sure the quadratic is 0
• if not add or subtract until all numbers and
  variables are on the same side.

60
Examples

• Solve each quadratic by factoring

61
Solve
62
Solving Quadratics by square roots

• When equations are in the form of y ax² you can
  just divide by a, then take the square root.
• You will have two answers

63
Solving Quadratic Equations
64
Example

• Smoke jumpers are firefighters who parachute into
  areas near forest fires. Jumpers are in free fall
  from the time they jump from a plane until they
  open their parachutes. The function y -16x²1600
  gives jumpers height y in feet after x seconds
  for a jump from 1600 ft. How long is the free
  fall if the parachute opens at 1000 ft?

65
Another way to solve

• Graphing is another way to solve quadratics.
• The solutions would be at the x intercepts of the
  parabola

66
Solve by graphing

• The last example and

67
5-5 Wrap Up

• Describe how the zero product property can be
  used to solve quadratic equations and which
  method of solving quadratics do you prefer? Why?

68
Warm Up 5-6

• Think about the square root of a negative number
• How do you think you could write the square root
  of a negative number? What would the square root
  mean?
• Please be creative with your responses

69
5-6 Complex Numbers

• OBJ Identify and graph complex numbers
• Add, subtract, and multiply complex numbers

70
(No Transcript)
71
Identifying Complex Numbers

• The system you use now is called the real number
  system
• Real number system is the rational, irrational,
  integers, whole, and natural numbers
• We will now expand our knowledge to include
  numbers like v-2

72
Examples
73
Simplify
74

• The imaginary number i is defined as the
  principal square root of -1
• iv-1, and i²-1
• Other imaginary numbers include -5i, iv2, and
  23i
• Numbers in the form abi form are called Complex
  Numbers

75
ABi (Complex Numbers)

• All real numbers are complex numbers where b0
• 50i5
• An imaginary number is also in the form abi, but
  b?0
• 05i5i

76
(No Transcript)
77
Simplify each number
78
Graphing Complex Numbers

• They are graphed like regular points
• The x coordinate is the real number part
• The y coordinate is the imaginary number part
• 35i would have the point (3,5)

79
Recall

• The absolute value of a real number is its
  distance from zero on a number line
• The absolute value of a complex number is its
  distance from the origin on the complex number
  plane

80
Formula to find distance
81
Find

• 5i
• 3-4i
• -3i
• 86i

82
Operations with Complex Numbers

• To add or subtract complex numbers, combine the
  real parts and the imaginary parts separately

83
Simplify the following expressions

• (57i)(-26i)
• (83i)-(24i)

84
Operations with Complex Numbers

• (3 4i) (7 8i) 2i(8 5i)
• (6-5i)(3 4i)
  (59i)(2-7i)
• (1i)(1-i)

85

• Multiply (5i)(-4i)
• Multiply (23i)(-35i)
• Simplify (3-2i)(-24i)
• (6-5i)(4-3i)
• (4-9i)(49i)

86
Solving quadratic equations using complex numbers

• Solve 4k²1000
• 3t²480
• 5x²-150

87
Wrap Up 5-6

• Describe the parts of a complex number and
  explain what they represent.

88
Warm Up 5-7

• (x7)²
• (x7)(x7)
• x²14x49
• How can you determine that this is equivalent to
  (x7)²

89
5-7 Completing the Square

• OBJ Solving quadratic equations by completing
  the square
• Rewriting quadratic equations in vertex form

90
Completing the Square
91
Rewrite the following equations and state the
vertex.
92
Solve the following

• x²8x-36
• x²-4x-8
• 5x²6x8

93

• A local florist is deciding how much money to
  spend on advertising. The function p(d)2000
  400d-2d² models the profit that the store will
  makes as a function of the amount of money it
  spends. How much should the store spend on
  advertising to maximize its profit?

94
Real World Example

• Suppose a ball is thrown straight up form a
  height of 4 feet with an initial velocity of 50
  feet per second. What is the maximum height of
  the ball?

95
Wrap Up 5-7

• Explain how to solve quadratic equation by
  completing the square.

96
Warm Up 5-8

• Given 4x²2x30
• What are the values of a, b, and c
• Find b, b², 4ac,
• b²-4ac
• v b²-4ac
• 2a
• -b v b²-4ac, -b- v b²-4ac

97
5-8 The quadratic formula

• OBJ Solving quadratic equations using the
  quadratic formula
• Determine types of solutions using the
  discriminant.

98
(No Transcript)
99
What is a Quadratic Equation

• A quadratic equation is an equation that can be
  written in the form

100
Quadratic Formula

• You can use the quadratic formula to calculate x
  using a, b, and c. YOU SHOULD MEMORIZE THIS
  FORMULA

101
Solve
102
Solve

• 2x² -6x -7
• 2x²4x-3

103

• Solve 10x2-13x-3 0
• Accounting for a drivers reaction time, the
  minimal distance in feet it takes for a certain
  car to stop is approximated by the formula
  d.042s2 1.1s4, where s is the speed in miles
  per hour. If a car took 200 feet to stop, about
  how fast was it traveling?

104
Discriminant Property

• Has 2 real solutions
• Has one real solution
• Has 2 complex
• solution

105
Without solving determine how many real solutions
the equations have
106

• The amount of power watts generated by a certain
  electric motor is molded by the equation
  P(l)120l-5l² where l is the amount of current
  passing through the motor in amperes (A). How
  much current should you apply to the motor to
  produce 600 W of power?

107

• A scoop is a field hockey pass that propels the
  ball from the ground into the air. Suppose a
  player makes a scoop that releases the ball with
  an upward velocity of 34 ft/sec. The function h
  -16t²34t models the height h in feet of the ball
  at time t in seconds. Will the ball ever reach a
  height of 20ft? If so how many seconds will it
  take? Will it reach 15 ft? How long will it take?

108
Lets use the graphing calculators

• x²6x80
• x²6x90
• x²6x100

109
A Way to Sum it Up
110
Wrap Up 5-8

• Describe how to use the quadratic formula to
  solve quadratic equations.