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Name______________________________ Mr. Rives – Algebra 2X
Unit 6: Radical Functions and Rational Exponents
DAY
1
2
3
4
TOPIC
8.6 LAWS OF EXPONENTS
RATIONAL EXPONENTS
SIMPLIFYING EXPRESSIONS
MORE 8.6
8.7 RADICAL FUNCTIONS (mini-quiz)
GRAPHS AND APPLICATION
8.8 SOLVING RADICAL EQUATIONS INCLUDING 2 RADICALS
5
MORE 8.8 / review
6
Quiz on days 1-5
9.4 OPERATIONS AND COMPOSITIONS OF
FUNCTIONS
7.2 INVERSES OF RELATIONS AND
FUNCTIONS
7
8
9
9.5 FUNCTIONS AND THEIR INVERSES
10
11
REVIEW
Quiz On days 7-9
ASSIGNMENT
Page 614 # 5-27 and 31-55
odd
WORKSHEET Day 2
Page 624 # 8-10,24-26, 3032, 39, 51-55a
Page 632 # 2-22
Page 633 # 27-41, page 635
#71-73
PAGE 686 # 15-17, 24-32,
39,40, 45-47
PAGE 501 # 1-16, 30, 34,
41-46
PAGE 693 # 9-19 ALL, 2435 ODD, 47-50 ALL
REVIEW SHEET
Page 1 of 18
Day 1: Laws of Exponents, Rational Exponents & Simplifying
Vocabulary:
a) Radical Expression:
b) Rational Exponent:
A radical expression can be written as a rational exponent, and vice versa. Here’s how…
3
4
5
y 
x
Properties of Exponents (Rational or Not)
Property in Words
Algebra
“Normal” Exponents
Product of Powers
am  an 
32  33 
Quotient of Powers
am

an
57

52
Rational Exponents
1
27
27
Power of a Power
a 

4 
Power of a Product
 ab 

 3x 
Power of a Quotient
a
  
b
m n
m
m
3 2
3
2
3
32  32 
2
3
1
3

4

 12 
4  
 

 25  36 2 
3
  
4
1
1
 27  3
  
 64 
EXAMPLES:
Page 2 of 18
Sometimes it is easier to convert radical expression into rational exponents before simplifying.
1)
4
58
2)
 x
10
5
3)

4
25

2
Other times we will need to simplify radical expressions.
x
even
Signs of Exponents and Radicals
odd
 x even
 x odd
x
even
x
odd
even x
x
odd x
To break down radicals, you must look at the ______________________.
Give the index: a)
3
x 2 yz
______
b)
4
x5 y10 z
______
#
Whatever the index is, that’s how large of a “group” that you need to bring an item out of the radicand.
Examples: (you should have done this before in algebra 1 and especially in finite math)
a)
3
8x8 y 2 z 6
index: ______
b)
75a11b4c7
index: ______
For this unit, we will be interested in simplifying rational exponents – in many cases we use the rules from
the previous page, but in some cases we will use the process for simplifying radicals.
Page 3 of 18
Day 2: HOMEWORK WORKSHEET
(classtime: game or activity)
Page 4 of 18
Day 3: Radical Functions – Graphs & Applications
Warm up: Continue the list of perfect squares as high as you can go…
x
x2
1
1
2
4
3
4
5
6
7
8
9
10
11
12
13
14
Today we are going to look at graphing square roots like y  x  3 (knowing perfect squares will help).
The parent function is __________________.
The general equation is y  a x  h  k . (slider demonstration)
a:
h:
k:
There are 3 ways to graph a square root: ___________________, ___________________, & table of values.
Method 1: Translations (best when the a value = ___)
a) y  x  3  2
b) y  x  5
c) y  x  8
Starting point: __________
Starting point: __________
Starting point: __________
Page 5 of 18
Method 2: Graphing Calculator (useful when a  1 , or anytime you are stuck on a graphing problem!)
Notes:
a) y  3 x  1
b) y  x  1  4
c) y  x  2
Method 3: Table of Values (useful anytime, and can be combined with either of the other 2)
a) y 
1
x 3
2
x
y
b) y   x  2  1
x
y
c) y  x  6
x
y
Page 6 of 18
Working backwards: Writing the equation when given a translation.
a) The parent function y  x is translated 2 units to the left and one unit down.
b) The parent function y  x is translated 3 units to the right.
c) The parent function y  x is compressed vertically by a factor of
1
and then translated 2 units up.
2
Individual Practice: Choose your method, but try some of each.
1) y  x  3
2) y  2 x
3) y  3 x  2  1
4) y  x  6
5) y  x  2
6) y  0.5 x  1
Closure: Describe the effect a, h, and k have on the equation y  a x  h  k
Page 7 of 18
Day 4: Solving Radical Equations – Including 2 Radicals
Steps to Solve:
1. Isolate the radical if it is not already: Move terms first, and then divide by any coefficient of the radical.
2. Square both sides of the equation.
3. Solve the remaining equation using approaches you already know.
4. CHECK ALL POSSIBLE SOLUTIONS IN THE ORIGINAL EQUATION TO BE CERTAIN THEY
ACTUALLY GIVE YOU A TRUE STATEMENT! FALSE ANSWERS ARE ______________________!
1.
x 1  7
2. 5 x  7  25
3.
x 8  4  x
4. 10  2 3x  1  14
x 1  4
6.
5.
3
3x  1  2 x  4 hmmmmmmm!?! What to do?
Page 8 of 18
What if you have no radical, but you do have a rational exponent? How do you undo it?
Remember that a square root is the same as raising to the ½ power! Wouldn’t we square (raise to the 2nd
power) to undo the square root if we have to get to the variable? So, …
RAISE BOTH SIDES TO THE ___________________ OF THE POWER YOU ARE TRYING TO UNDO!
5
7. x 6  32
3
2
9. ( x  4)  8
2
8. (2 x  5) 3  16
1
2
10. (4 x  8)  2 x
_________________________________________________________________________________
11. Ok, now. What should we do with this one?
2x  3  3  2x
Page 9 of 18
Day 5: Activity Practicing Yesterday’s Lesson
Day 6: Quiz on Day’s 1-5
Day 7: Operations and Compositions of Functions
Function Operations
Operation
Function Operation
( f  g )( x)  f ( x)  g ( x)
Example in Terms of x
f  x   5x2  4 x
Example with numbers
( f  g )(3) 
g  x   5x  1
Addition
( f  g )( x)  f ( x)  g ( x)
f  x   2 x 2  3x  1
( f  g )(2) 
g  x   5x  1
Subtraction
( f  g )  x   f ( x)  g ( x )
f  x   x2  3
( f  g )  4  
g  x  x  5
Multiplication
f 
f ( x)
, g ( x)  0
  ( x) 
g ( x)
g
f  x   x2 1
g  x  x 1
 f 
  (6) 
g
Examples on next page!
Examples on next page!
Division
Domain Restriction!
Composition
f
g  x   f  g  x  
g
f  x   g  f  x  
Page 10 of 18
Method 1: Finding the composition of a number.
f  x   5x and g  x   x2  1
Find  f g  3  _________________ .
Work Inside – Out  So first find___________ = _____
g  x   x2  1
Try on your own:
f  x   3x  2
g  x   2x2
 Next, find g  ______  
f  x   5x
find f  g  2   .
Method 2: Find the composition of a variable.
f  x   5x and g  x   x2  1
Find f  g  x   .
Work Outside – In  Write out f first 
Now, plug all of g: (
) in for x 
Simplify 
What if we were trying to find g  f  x   ?
Note: If you were trying to find the composition of a number, you could use this method first and then plug
in the number into your answer.
Page 11 of 18
Additional Practice:
Page 12 of 18
Day 8: Inverses of Relations & Functions
A ______________________ is a pairing of two values, normally in the form  x, y  .
Plot each ordered pair given below. Then, write the inverse point by switching  x, y  to  y, x  .
Point  x, y 
Inverse  y, x 
1,0
 3,1
 4, 2 
5,3
 6,5
Important: What line is each point reflected over?
Inverses are created by switching x with y. Let’s find the inverse of some equations…
a) y  5 x  2
d) f  x  
1
 3x  1
2
b) y 
x
4
3
e) f  x  
c) f  x   5x
2x  5
3
Always change f  x 
to _____ first!
Directions: Find the inverse, then graph both the function and inverse
f  x 
x
2
3
Page 13 of 18
Question #1: What determines if a relation is a function or not?!
Questions #2: What are the domain and range of a relation (or function)?
Directions: Graph the relation and connect the points. Then graph the inverse. Identify the domain and
range of each, and identify if either (or both) are functions or just relations.
RELATION
INVERSE
x
-1
2
3
5
x
y
1
3
5
5
y
Domain:
Domain:
Range:
Range:
Function?:
Function?:
Graph the line that passes through the points  2, 3 and  8,1 .
Next, graph the line through the inverse points.
Finally, find the slope of each of the two lines.
What’s the connection?
Additional Practice/Wrap Up: Find each inverse.
1) f  x   15x 10
2) f  x  
x  12
4
3) f  x   5x  2
Page 14 of 18
Day 9: Functions and their Inverses
f  x  4x
Warm up: Are the following functions inverses? Explain how you know.
g  x 
1
x
4
There is another way to prove that two functions are inverses: By using ____________________ functions.
Let’s find f  g  x   
g  f  x   
and
When BOTH of these functions = ______, that means that the functions are inverses of each other!
Example #2: Determine if the following functions are inverses by using composition functions.
f  x 
1 2
x
4
and
g  x  2 x
The graph of f  x   x2  3 is shown.
First, graph the inverse by using the line of symmetry.
Next, find the inverse algebraically, and graph it
to check your graph of the inverse.
Is the inverse a function, or just a relation?
There is a trick to find if the INVERSE of a function will be a function without even finding the inverse.
This is known as the _______________________ line test (like the vertical line test, but horizontal!)
a)
b)
Page 15 of 18
Additional Practice:
Page 16 of 18
Day 10: Review for Unit 6 Quiz
Given f ( x)  2 x  4 , g ( x)  x 2  9 , and h( x)  x  5 , find the following:
1. f (7) 
7. f 1 ( x) 
2. g (0) 
8. ( fg )( x) 
(multiply)
3. h(10) 
9. ( g  f )( x) 
4. f ( g (4)) 
compositions
10. f ( f 1 ( x)) 
5. h( f (14)) 
11. h( g ( x )) 
6. g (h( f (2))) 
(double composition)
12. g ( f ( x)) 
13. Determine using composition if j ( x)  6 x  1 and k ( x)  6 x  1 are inverses. Explain how you know.
Page 17 of 18
14. Refer to the graph.
Is the given graph a function? How do you know?
Is the inverse of the given graph a function? How do you
know?
Draw the graph of the inverse on the same axes.
15. The points (9, 13) and (-4, 10) are on p ( x ) . Name 2 points on p 1 ( x) .
16. Is it always true that f ( g ( x))  g ( f ( x)) ? If yes, state why. If no, give an example where it’s not true.
1.) 10
2.) -9
3.) 5
4.) 10
5.) 29
6.) 16
12.) g ( f ( x))  4 x 2  16 x  7
13.) They are not, since j (k ( x))  x .
14.) The graph is not (fails VLT), but its inverse is (the
graph passes HLT).
x4
2
8.) ( fg )( x)  2 x3  4 x 2  18x  36
7.) f 1 ( x) 
9.) ( g  f )( x)  x 2  2 x  5
10.) f ( f 1 ( x))  x (composing a function and
its inverse always yields x!)
2
11.) h( g ( x))  x  4
15.) (13, 9) and (10, -4)
16.) NO – you can use f and g from the 1st set of problems
as your example!
Page 18 of 18