Day 40 March 30 th Objective Learn how

$Day 40: March 30 th Objective: Learn how to simplify algebraic fractions. THEN Understand$

Day 40: March 30 th Objective: Learn how to simplify algebraic fractions. THEN Understand how to multiply and divide rational expressions and continue to learn how to simplify rational expressions. • • • Homework Check Notes: Simplify Rational Expressions 1 W 2 (odds) Notes: Multiplying/Dividing Rational Expressions 2 W 2 (odds) Closure Homework: Finish EVENS from Classwork Project Due: Wednesday, April 6 th (Rubric)

Simplifying Rational Expressions Simplify the following expressions:

Simplifying Rational Expressions A fellow student simplifies the following expressions: Which simplification is correct? Substitute two values of x into each to justify your answer. MUST BE MUITLIPLICATION!

Simplifying Rational Expressions Simplify: Can NOT cancel since everything does not have a common factor and its not in factored form Factor Completely CAN cancel since the top and bottom have a common factor

Multiplying and Dividing Fractions Multiply Numerators Multiply: Divide: Multiply by the reciprocal (flip) Multiply Denominators Remember to Simplify!

Multiplying/Dividing Rational Expressions Simplify: Half the work is done! Combine Rewrite Cancel

Multiplying/Dividing Rational Expressions Simplify: Flip to turn it into a multiplication Factor Completely Cancel

Day 41: March 31 st Objective: Understand how to add and subtract rational expressions and continue to learn how to simplify rational expressions. • • • Homework Check Continue to Work on the first 2 worksheets Notes: Adding and Subtracting Rationals Rational Expressions 3 W 2 Closure Homework: Finish all worksheets Project Due: Wednesday, April 6 th (Rubric)

Adding and Subtracting Fractions Subtraction: Addition: Common Denominator Add the Numerators Least Common Denominator (if you can find it) Subtract the Numerators Remember to Simplify if Possible!

Add/Subtract Rational Expressions Simplify: Same denominator! Half the work is done! CAREFUL with subtraction! Combine Like Terms Make sure it can’t be simplified more

Add/Subtract Rational Expressions Simplify: Find a Common Denominator Combine Like Terms

Add/Subtract Rational Expressions Simplify: Find a common denominator Distribute numerators but leave the denominators factored CAREFUL with subtraction Combine like Terms

Add/Subtract Rational Expressions Simplify: Factor to find a Smaller Common Denominator Make sure it can’t be simplified beforehand

Add/Subtract Rational Expressions Simplify: Factor to find a Smaller Common Denominator Make sure it can’t be simplified more

Day 42: April 1 st Objective: Consider two functions and identify the relationships between the functions and the system from which they come. • Homework Check • Rational Expressions 4 W 2 • Wells Time • 5 -96 (pg 249, Rsrc. Pg) • Closure: Final Challenge Homework: Finish Worksheet AND 6 -8 to 6 -15 (pg 265 -266) Project Due: Wednesday, April 6 th (Rubric)

Day 43: April 4 th Objective: Learn to find rules that “undo” functions, and develop strategies to justify that each rule undoes the other. Also, graph functions along with their inverses and make observations about the relationships between the graphs. THEN Introduction to the term inverse to describe undo rules. Also graphing the inverse of a function by reflecting it across the line of symmetry and write equations for inverses. • Homework Check • 6 -1 to 6 -6 (pgs 263 -265) • Wells Time • START: 6 -16 to 6 -25 (pgs 267 -269, Rsrc. Pg) • Closure Homework: 6 -7 (pg 265) AND 6 -26 to 6 -32 (pgs 270) Project Due: Wednesday, April 6 th (Rubric)

Guess my Number I’m thinking of a number that… When I… I get… My number is… • Add four to my number AND • Multiply by ten -70 -11 • Double my number • Add four AND • Divide by two Five Three • Square my number • Add three • Divide by two AND • Add one Seven Three 3 and… -3 • Double my number • Subtract six • Take the square root Eleven Eight

Inverse Notation Original function Inverse function

“Undo” Rule 1 st Step p(x) Add 3 p -1 (x) Divide 2 2 nd Step Cube 3 rd Step Start Multiply 2 Cube Root Subtract 3 Only works when there is one x!

Tables and Graphs of Inverses Switch x and y Orginal (0, 25) (2, 16) (18, 16) (6, 4) (14, 4) (10, 0) Function X 0 2 6 10 14 18 20 Y 25 16 4 0 4 16 25 X 25 16 4 0 4 16 25 Y 0 2 6 10 14 18 20 Inverse Switch x and y (16, 18) (4, 14) (0, 10) (4, 6) (16, 2) Non-Function Line of Symmetry: y = x

6 -6: Learning Log Title: Finding and Checking Undo Rules • What strategies did your team use to find undo rules? • How can you be sure that the undo rules you found are correct? • What is another name for “undo? ” • How do the tables of a rule and an undo-rule compare? Graph?

Day 44: April 5 th Objective: Introduction to the term inverse to describe undo rules. Also graphing the inverse of a function by reflecting it across the line of symmetry and write equations for inverses. THEN Use the idea of switching x and y-values to learn how to find an inverse algebraically. Also learn about compositions of functions and use compositions f(g(x)) and g(f(x)) to test algebraically whether two functions are inverses of each other. • Homework Check • Finish: 6 -16 to 6 -25 (pgs 267 -269 , Rsrc. Pg) • Wells Time • 6 -38 to 6 -42 (pgs 272 -274) • Closure Homework: 6 -33 to 6 -37 (pgs 271) AND 6 -44 to 6 -53 (pgs 274277) Project Due: Wednesday, April 6 th (Rubric)

The Rule for an Inverse 1 st Step p(x) p -1 (x) Add 2 Add 6 2 nd Step Square Divide 3 3 rd Step 4 th Step Start Multiply 3 Subtract 6 ± Square Root Subtract 2

Vertical Line Test If a vertical line intersects a curve more than once, it is not a function. Use the vertical line test to decide which graphs are functions.

Horizontal Line Test If a horizontal line intersects a curve more than once, the inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function.

Restricted Domain Find the inverse relation of f below: Inverse Function

Day 45: April 6 th Objective: Use the idea of switching x and y-values to learn how to find an inverse algebraically. Also learn about compositions of functions and use compositions f(g(x)) and g(f(x)) to test algebraically whether two functions are inverses of each other. • Homework Check • 6 -38 to 6 -42 (pgs 272 -274) • Closure Homework: 6 -44 to 6 -53 (pgs 274 -277) Project Due Today

Algebraically Finding an Inverse Find the inverse of the following: Switch x and y Solve for y Do not write y-1 Make sure to check with a table and graph on the calculator.

Algebraically Finding an Inverse Find the inverse of the following: Switch x and y Solve for y 2 Because x =9 has two solutions: 3 & -3 Do not write y-1 Make sure to check with a table and graph on the calculator.

Algebraically Finding an Inverse Find the inverse of the following: Switch x and y Really y = Solve for y Make sure to check with a table and graph on the calculator.

Algebraically Finding an Inverse Only Half Parabola Find the inverse of the following: Switch x and y Really y = Solve for y Full Parabola (too much) Restrict the Domain! x=3 Make sure to check with a table and graph on the calculator.

Composition of Functions Substituting a function or it’s value into another function. Second g f First (inside parentheses always first) OR

Composition of Functions Let Our text uses the first one Plug x=1 into g(x) first and . Find: Equivalent Statements Plug the result into f(x) last

Composition of Functions Let Plug x into f(x) first and . Find: Plug the result into g(x) last

Inverse and Compositions In order for two functions to be inverses: AND

Day 46: April 7 th Objective: Apply strategies for finding inverses to parent graph equations. Begin to think of the inverse function for y=3 x. THEN Define the term logarithm as the inverse exponential function or, when y=bx, “y is the exponent to use with base b to get x. ” • Homework Check • 6 -54 to 6 -58 (pgs 277 -279) • Wells Time • 6 -67 to 6 -71 (pgs 281 -282) • Closure Homework: 6 -59 to 6 -66 (pgs 279 -280) AND 6 -72 to 6 -80 (pgs 283 -284) Project Due: Monday, April 4 th (Rubric)

Silent Board Game

Logarithm and Exponential Forms Logarithm Form 5 = log 2(32) Logs Give you Exponents Input Becomes Output Base Stays the Base 5 2 = 32 Exponential Form

Examples Write each equation in exponential form 1. log 125(25) = 2/3 1252/3 = 25 2. Log 8(x) = 1/3 81/3 = x Write each equation in logarithmic form 1. If 64 = 4 3 2. If 1/27 = 3 x log 4(64) = 3 Log 3(1/27) = x

Inverse of an Exponential Equation Original Inverse OR Log’s give you exponents!

Definition of Logarithm The logarithm base a of b is the exponent you put on a to get b: a>0 and b>0 i. e. Logs give you exponents!

6 -71: Closure 2 4 7 1. 2 w+3

Day 47: April 8 th Objective: Assess Chapters 1 -5 in an individual setting. • Homework Check • Midterm Exam • Closure Homework: 6 -84 to 6 -92 (pgs 286 -287)

Day 48: April 11 th Objective: Develop methods to graph logarithmic functions with different bases. Rewrite logarithmic equations as exponential equations and find inverses of logarithmic functions. THEN Look into the base of the log key on the calculator. Also extend our knowledge of general equations for parent functions to transform the graph of y=log(x). • Homework Check • Logarithms and Graphs Packet (Extra) • Wells Time • 6 -93 and 6 -95 (pgs 288 -289) • Closure Homework: 6 -96 to 6 -105 (pgs 290 -291)

6 -83: Learning Log Title: The Family of Logarithmic Functions • • • What is the general shape of the graph? What happens to the value of y as x increases? How is the graph related to the exponential graph? What is the Domain? Range? Why is the x-intercept always (1, 0)? Why is the line x=0 (y-axis) always an asymptote? Why is there no horizontal asymptote? How does the graph change if b changes? What does the graph look like when 0<b<1? What does the graph look like when b=1? What does the graph look like when b>1?

Common Logarithm Ten is the common base for logarithms, so log(x) is called a common logarithm and is shorthand for writing log 10(x). You read this as “the logarithm base 10 of x. ” Our calculator has the button log. It doesn’t have the subscript 10 because it stands for the common logarithm: log 10100 = log 100

Logarithmic Function Parent Equation Graphing Form

Example: Exponential Transformation: Shift the parent graph three units to the right and two units up. New Equation: y=2 x=3