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.. 5-1 Modeling Data with Quadratic Functions. OBJ: Recognize and use quadratic functions Decide whether to use a linear or a quadratic model. Quadratic Functions. Quadratic function is a function that can be written in the form f(x)= ax
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)= ax²+bx+c, 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)= (-x+3)(x-2)
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. Y=ax²+bx+c is the general equation of a parabola
If a>0 the parabola opens up
If a<0 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
Y=a(x-h)²+k is the vertex form of a parabola
It is a translation of y=ax²
(h,k) are the coordinates of the vertex
18. Sketch the following graphs
19. The vertex is ( h, k)
Axis of symmetry is x = h
If a > 0 it is a max
If a < 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)²+3
Sketch the graph of y = 3(x+1)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 y=ax²+bx+c it is standard form
The x coordinate of the vertex can be found by –b/(2a)
To find the y coordinate by – [(b^2-4ac)/4a]
27. Suppose a toy rocket is launched to its height in meters after t seconds is given by H = -4.9t^2 +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= 2x²+10x+7 in vertex form
Write the function y= -x²+3x-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)²+12 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.
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
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)
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 y=x
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 y=x^2
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 ab=0, then a=0 or b=0
(x+3)=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
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= -16x²+1600 gives jumper’s 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
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
74. The imaginary number i is defined as the principal square root of -1
i=v-1, and i²=-1
Other imaginary numbers include -5i, iv2, and 2+3i
Numbers in the form a+bi form are called Complex Numbers
75. A+Bi (Complex Numbers) All real numbers are complex numbers where b=0
An imaginary number is also in the form a+bi, but b?0
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
3+5i 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|
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 (5+7i)+(-2+6i)
84. Operations with Complex Numbers (3 +4i) + (7 + 8i) 2i(8 + 5i)
(6-5i)+(3 +4i) (5+9i)(2-7i)
85. Multiply (5i)(-4i)
86. Solving quadratic equations using complex numbers Solve 4k²+100=0
87. Wrap Up 5-6 Describe the parts of a complex number and explain what they represent.
88. Warm Up 5-7 (x+7)²
How can you determine that this is equivalent to (x+7)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: x²=8x-36
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 4x²+2x+3=0
What are the values of a, b, and c
Find –b, b², 4ac,
-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.
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
102. Solve 2x²= -6x -7
103. Solve 10x^2-13x-3 =0
Accounting for a driver’s reaction time, the minimal distance in feet it takes for a certain car to stop is approximated by the formula d=.042s^2 + 1.1s+4, 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
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 = -16t²+34t 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 x²+6x+8=0
109. A Way to Sum it Up
110. Wrap Up 5-8 Describe how to use the quadratic formula to solve quadratic equations.