1. Unit 8: Radical & Rational Functions
LG 8-1 Radical Functions
LG 8-2 Rational Functions
TEST 5/22 – the last day of school!
remediation

2. LG 8-1 Radical Functions We will SOLVE them! 2) We will GRAPH them!
Understand solving equations as a process of reasoning and explain the reasoning
MGSE9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
2) We will GRAPH them!
Analyze functions using different representations
MGSE9-12.F.IF.7 Graph square root, cube root functions expressed algebraically and show key features of the graph both by hand and by using technology.
MGSE9-12.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior;
3) We will ANALYZE them!
Create equations that describe numbers or relationships
MGSE9-12.A.CED.2 Create radical functions in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Interpret functions that arise in applications in terms of the context
MGSE9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

3. ENDURING UNDERSTANDINGS
Graph radical functions
Solve radical equations algebraically and graphically
ESSENTIAL QUESTIONS
How can we extend arithmetic properties and processes to algebraic expressions and how can we use these properties and processes to solve problems?
Why are all solutions not necessarily the solution to an equation? How can you identify these extra solutions?

4. SELECTED TERMS AND SYMBOLS
Extraneous Solutions: A solution of the simplified form of the equation that does not satisfy the original equation.
 Inequality: Any mathematical sentence that contains the symbols > (greater than), < (less than), < (less than or equal to), or > (greater than or equal to).

5. EVIDENCE OF LEARNING
By the conclusion of this unit, you should be able to demonstrate the following competencies:
Solve radical equations
Graph radical functions and identify key characteristics
Interpret solutions to graphs and equations given the context of the problem

6. Solving radical equations
A radical equation contains a variable within a radical.
Recall that you can solve quadratic equations by taking the square root of both sides. Similarly, radical equations can be solved by raising both sides to a power.

7. Example 1: Solving Equations Containing One Radical

8. Example 2: Solving Equations Containing One Radical

9. Example 3: Solving Equations Containing Two Radicals

10. You Try! Example 4

11. Extraneous Solutions
Raising each side of an equation to an even power may introduce extraneous solutions.
You don’t have to worry about extraneous solutions when solving problems to an odd power.

12. Example 5

13. You Try! Example 6

14. Example 7

15. Example 8: Solving Equations with Rational Exponents1 3
(5x + 7) = 3

16. Example 9: Solving Equations with Rational Exponents
2x = (4x + 8) 1 2

17. Example 10 1 2
3(x + 6) = 9

18. TOTD On a sticky note, please write your name and your answers to the following questions:

19. Class work/Homework:
Page 462 #1 – 22

20. Warm Up 5/11

21. Graph of the Square Rootx y -1 i 1 4 2
Square Root – “SHOOT”
Note: We cannot graph imaginary numbers on the coordinate plane. Therefore, the graph stops at x = 0.

22. “Free-Style Swim” Graph of the Cube Root x y -4 -1 1 4 1.59 -1.591 4
1.59
“Free-Style Swim”
Note: Since the index number is odd, we can graph the function for all x values. Therefore, the domain is all reals.

23. Inverses
Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs below show the inverses of the quadratic parent function and cubic parent function.

24. Radical Functions - Transformations

25. Radical Functions - Transformations
When “a” is negative: Reflect over the x-axis
When “a” is a fraction between 0 and 1: Vertical Shrink (Compression)
When “a” is a number greater than 1: Vertical Stretch
**Negative on the outside – it “x-caped”**

26. Examples: Reflect over the x-axis Vertical Stretch by 2
Reflect over the x-axis, Vertical Shrink by 1/3

27. Radical Functions - Transformations
**Inside the radical, opposite of what you think**
When “b” is negative: Reflect over the y-axis
When “b” is a fraction between 0 and1: Horizontal Stretch
When “b” is a number greater than 1: Horizontal Shrink (Compression)
**Negative on the inside – “y” am I in here?**

28. Examples: Reflect over the y-axis Horizontal Shrink by 1/3
Horizontal Stretch by 4

29. Radical Functions - Transformations
**Inside the radical, opposite of what you think**
If “h” is positive, then the graph moves left: Horizontal shift to the left
If “h” is negative, then the graph moves right: Horizontal shift to the right

30. Radical Functions - Transformations
If “k” is positive, then the graph moves up: Vertical shift up
If “k” is negative, then the graph moves down: Vertical shift down

31. Examples:
Right 3
Down 5
Left 2, Up 4

32. Going Backwards:
The parent function is reflected across the x-axis, stretched vertically by a factor of 4, and translated 1 unit up

33. Going Backwards:
The parent function is stretched horizontally by a factor of 2, reflected across the y-axis, and translated 3 units left

34. Use your table to help find good points
Radical Functions - Graphing
Use your table to help find good points
“Starting point” will always be (h, k)

35. Radical Functions – Let’s Graph!!

36. Radical Functions

37. Describe how to obtain the graph of g from the graph of f.
Vertical shift:
3 units up
Horizontal shift:
5 units to the left
Vertical shift:
4 units down
Horizontal shift:
1 units to the right
Vertical shift:
6 units up
Horizontal shift:
1 units to the left

38. Analyzing Graphs of Radical Functions
Graph the radical function and identify the characteristics:
Stretch
4 times as fast
Domain:
Range:
Interval of increase/decrease:
x-int:

39. Graph the radical function and identify the characteristics:
Stretch
2 times as fast
Horizontal shift:
3 units to the left
Domain:
Interval of increase/decrease:
Range:
x-int:

40. Graph the radical function and identify the characteristics:
Stretch
3 times as fast
Horizontal shift:
2 units to the right
Interval of increase/decrease:
Domain:
Range:
x-int:

41. Graph the radical function and identify the characteristics:
Reflect over x-axis
Stretch 2 times as fast
Horizontal shift:
Vertical shift:
1 unit to the left
2 units down
Domain:
Interval of increase/decrease:
Range:
x-int:

42. Graph the radical function and identify the characteristics:
Reflect over y-axis
Stretch 2 times as fast
Interval of increase/decrease:
Domain:
Range:
x-int:

43. Graph the radical function and identify the characteristics:
Reflect over x-axis
Horizontal shift:
Vertical shift:
2 units to the right
1 unit up
Domain:
Interval of increase/decrease:
Range:
x-int:

44. TOTD On a sticky note, please write your name and your answers to the following 3 questions:

46. Homework pg 454 #

47. Warm UP 5/12
Describe the transformations of the graph of each function. Then, state the domain, range, and intervals of increase or decrease.

48. Homework Check Silently answer the following questions on a sticky note and submit them. Don’t forget your name.

49. Comparing Functions

50. Comparing Functions

51. Comparing Functions

52. Comparing Functions

53. Comparing Functions

54. Comparing Functions

55. Comparing Functions

56. Comparing Functions

57. TOTD

58. Homework: Read pg 460 Example 5
Homework: Read pg 460 Example 5. Then, complete #23 – 25 on pg 462 and #42 – 44 on pg 463

59. Warm UP 5/15

60. Applications of Radical Functions
The Pendulum Problem Gabriel visited the Museum of History and Technology with his physics class. They saw Focault’s Pendulum and it was fascinating to Gabe. He knew from physics class that the time it takes a pendulum to complete a full cycle or swing depends upon the length of the pendulum. The formula is given by where T represents the time in seconds and L represents the length of the pendulum in feet. He timed the swing of the pendulum with his watch and found that it took about 8 seconds for the pendulum to complete a full cycle. Help him figure out the length of the pendulum in feet.
𝟓𝟐 𝒇𝒆𝒆𝒕

61. The Pendulum Problem
Jimmy thought that a pendulum that took a full 20 seconds to complete a full cycle would be very dramatic for a museum. How long must that pendulum be? If ceilings in the museum are about 20 feet high, would this pendulum be possible?
𝟑𝟐𝟒 𝒇𝒆𝒆𝒕; The building would have to be over 16 stories tall to accommodate this pendulum!

62. Classwork
Radical Characteristics Practice Worksheet

63. QUUEEEEEZZZZZ

64. Warm Up 5/16 F

65. LG 8-2 Rational Functions
Rewrite rational expressions
MGSE9-12.A.APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Create equations that describe numbers or relationships
MGSE9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only).
MGSE9-12.A.CED.2 Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase “in two or more variables” refers to formulas like the compound interest formula, in which A = P(1 + r/n)nt has multiple variables.) (Limit to radical and rational functions.)
Understand solving equations as a process of reasoning and explain the reasoning
MGSE9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Interpret functions that arise in applications in terms of the context
MGSE9-12.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (Limit to radical and rational functions.) 
MGSE9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (Limit to radical and rational functions.)
Analyze functions using different representations
MGSE9-12.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. (Limit to radical and rational functions.)
MGSE9-12.F.IF.7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

66. ENDURING UNDERSTANDINGS
Recognize rational functions as the division of two polynomial functions and rewrite a rational expression
Find the sum, difference, product, and quotient of rational expressions
Graph rational functions
Interpret graphs and discover characteristics of rational functions
Solve rational equations algebraically and graphically
Solve rational inequalities

67. ESSENTIAL QUESTIONS
How can we extend arithmetic properties and processes to algebraic expressions and how can we use these properties and processes to solve problems?
How do the polynomial pieces of a rational function affect the characteristics of the function itself?
How are horizontal asymptotes, slant asymptotes, and vertical asymptotes alike and different?
Why are all solutions not necessarily the solution to an equation? How can you identify these extra solutions?
Why is it important to set a rational inequality to 0 before solving?

68. SELECTED TERMS AND SYMBOLS
Extraneous Solutions: A solution of the simplified form of the equation that does not satisfy the original equation.
 Inequality: Any mathematical sentence that contains the symbols > (greater than), < (less than), < (less than or equal to), or > (greater than or equal to).
Rational Function: The quotient of two polynomials
Reciprocal: Two numbers whose product is one.

69. EVIDENCE OF LEARNING
By the conclusion of this unit, you should be able to demonstrate the following competencies:
Rewrite rational expressions in different forms
Add, subtract, multiply, and divide rational expressions
Solve rational equations
Solve rational inequalities
Graph rational functions and identify key characteristics
Interpret solutions to graphs and equations given the context of the problem

70. Warm UP 5/17
Solve this equation

71. Warm UP
Describe the characteristics of f(x).

72. And more characteristics!
Graphing Rational Functions
. . . (by hand)
And more characteristics!

73. Behavior with asymptotes

74. Describe the graph of

75. Describe the graph of

76. Describe the graph of

77. Warm UP!
Describe the graph of

78. About those holes…
Sometimes, a factor will appear in the numerator and in the denominator. Let's assume the factor (x-k) is in the numerator and denominator. Because the factor is in the denominator, x=k will not be in the domain of the function. This means that one of two things can happen. There will either be a vertical asymptote at x=k, or there will be a hole at x=k.
𝑓 𝑥 = 𝑥 2 +3𝑥−10 𝑥 2 +8𝑥+15
𝑓 𝑥 = 𝑥 2 −2𝑥+1 𝑥 2 +2𝑥−3
𝑓 𝑥 = 𝑥 2 +2𝑥−3 𝑥 2 −2𝑥+1

79. Let's look at what will happen in each of these cases.
There are more (x-k) factors in the denominator. After dividing out all duplicate factors, the (x-k) is still in the denominator. Factors in the denominator result in vertical asymptotes. Therefore, there will be a vertical asymptote at x=k.
𝑓 𝑥 = 𝑥 2 +2𝑥−3 𝑥 2 −2𝑥+1

80. Let's look at what will happen in each of these cases.
There are more (x-k) factors in the numerator. After dividing out all the duplicate factors, the (x-k) is still in the numerator. Factors in the numerator result in x-intercepts. But, because you can't use x=k, there will be a hole in the graph on the x-axis.
𝑓 𝑥 = 𝑥 2 −2𝑥+1 𝑥 2 +2𝑥−3

81. Let's look at what will happen in each of these cases.
There are equal numbers of (x-k) factors in the numerator and denominator. After dividing out all the factors (because there are equal amounts), there is no (x-k) left at all. Because there is no (x-k) in the denominator, there is no vertical asymptote at x=k. Because there is no (x-k) in the numerator, there is no x-intercept at x=k. There is just a hole in the graph, someplace other than on the x-axis. To find the exact location, plug in x=k into the reduced function (you can't plug it into the original, it's undefined, there), and see what y-value you get.
𝑓 𝑥 = 𝑥 2 +3𝑥−10 𝑥 2 +8𝑥+15

82. Describe the graph of

83. Sketch the graph of

84. Steps to describing and graphing (by hand):
Factor numerator and denominator
Locate any holes (if applicable) and plot
Sketch the vertical asymptote(s)
Sketch the HA or SA and find intersections with asymptotes
Plot x and y intercepts
Make a sign chart
Connect the “dots”!

85. DO THEM ALL EVEN IF YOU DON’T WANT TO DO THEM BECAUSE I SAID SO
Practice – 8 graphs
DO THEM ALL EVEN IF YOU DON’T WANT TO DO THEM BECAUSE I SAID SO

86. Rational Function Characteristics and Graphs
Describe & Graph

87. Describe & Graph

88. Describe & Graph

89. Describe & Graph

90. Describe & Graph

91. Describe & Graph

92. Describe & Graph

93. Describe & Graph

94. Warm UP!
Sketch the graph WITHOUT USING A CALCULATOR! Then determine the characteristics: Domain: Range: End Behavior: Intervals of Increase/Decrease:

96. Solving Equations Containing
Rational Expressions

97. First, we will look at solving these problems algebraically.
Here is an example that we will do together using two different methods.

98. Eliminate the fractions
The best way to solve a rational equation:
Eliminate the fractions
This can be done by multiplying each side of the equation by the LCD.

99. (x+2)(x-5)
What is the LCD?

100. It is VERY important that you check your answers!!!!!!!!!!!!!!!!!!

101. The other method of solving rational equations is cross-multiplication.

102. Here is another example that we will do together:

103. Step 1: Find the LCD Hint: Factor the denominator Therefore….
This denominator can be factored into 3(x-2)
Therefore….

104. Step 2: Multiply both sides of equation by LCD
Step 2: Multiply both sides of equation by LCD. This eliminates the fraction.

105. Step 3: Solve for x

106. Since there are two answers, there needs to be two checks.
Let x =

107. Check #2:
Let x = 2
When you check the number 2, you get a zero in the denominator. This means that 2 can not be a solution.

108. Now, you do these on your own.

109. Example #3:
A car travels 500 miles in the same time that a train travels 300 miles. The speed of the car is 30 miles per hour faster than the speed of the train. Find the speed of the car and the train.

110. Remember the formula d=rt where:
r = rate of speed
d = distance
t = time
Since both vehicles travel the same amount of time, solve the formula for t.

111. Identify the variables that you are going to use.
Let r = speed of the train
How do you represent the speed of the car?
Let r+30 = speed of the car

112. Car’s time =
Train’s time

113. How would you solve this equation?
Cross-multiply

114. Make sure that you answer the question.
The car travels at a speed of 75mph
The train travels at a speed of 45 mph