Fundamental Principle of Counting: If event A can occur in m ways and event B can occur in n ways, then both events can occur in m n ways.

Transcription

1 ELM Tutorial Unit.1 Apply the Counting Principle Fundamental Principle of Counting: If event A can occur in m ways and event B can occur in n ways, then both events can occur in m n ways. Suppose there are roads from Aldridge to Boonsville and roads from Boonsville to Casper, in how many ways can one travel from Aldridge to Casper through Boonsville? Using the fundamental counting principle, there are. = 6 ways to travel from Aldridge to Casper. Eample 1: The silver maple tree grows branches per month and each branch has 0 leaves. In 1 months how many new leaves does the tree acquire? Eample : A shirt company has 4 designs that can be made with short or long sleeves. There are color patterns available. How many different types of shirts are available from this company? Try These 1: The silver maple tree grows branches per month and each branch has 40 leaves. How many months would it take to acquire 1600 new leaves? Try These : A shoe store stocks styles of shoes. Each style has 4 colors and 6 sizes. How many pairs of shoes must the store carry to have a pair of each combination? 006 Colorado State University Mathematics Department 1

2 ELM Tutorial Unit. Understanding the Terminology of Algebra A constant represents a quantity that does not change. It may be a numeral or a symbol whose value is taken to be a constant. For eample:, 4.8, ½, π (pi), or c (the speed of light). A variable is a symbol (usually a letter from the alphabet) used to represent a value from a particular set (usually the set of real numbers). The term coefficient is used to signify something that multiplies something else. For instance, in 4y z, 4y is the coefficient on z since that is what is being multiplied by z. Usually, a person is specifically interested in the numerical coefficient. In this case, 4 is the numerical coefficient on the variable portion y z. Note: the coefficient of a single variable, such as, is 1. Variables and constants can be combined with the operations of arithmetic to produce algebraic 6 y 9y epressions. For eample: + and. 16y Algebraic terms are part of an algebraic sum. Therefore, terms in an algebraic epression are separated by addition or subtraction signs. For eample, 4y z is one term, the epression + contains two 6 y 9y terms, and, the numerator of the epression contains the terms 6 y and9 y. 16y Two terms are like terms if all parts of both terms ecept the numerical coefficients are the same. For eample, a b c 4 and -6. a b c 4 are like terms while y and y are not like terms. To combine like terms, we simply combine the coefficients making use of the distributive property. For eample, a b c 4 6. a b c 4 = ( 6.) a b c 4 = -. a b c 4. Eample 1: Which of the following are like terms? a) 4 y b) 4y c) y d) - y e) y Eample : Which of the following are terms of the epression 4a b 1ab + ab + 8? a) 4a b b) ab c) a b d) 1 e) 8 Try These 1: Which of the following are constants? a). b) π c) y d) -7 e) Try These : Which of the following are variables in the following epression 4a b 1ab + ab + 8? a) -1 b) 4 c) b d) 8 e) a 006 Colorado State University Mathematics Department

3 ELM Tutorial Unit. Simplify Algebraic Epressions Order of Operations Parentheses (simplify within any parentheses) Eponents (calculate any powers) Multiplication and Division (do as they appear from left to right) Addition and Subtraction (do as they appear from left to right) Simplify the epression: 0 [ ( )] + 0 [ ( )] [ ( 9 1) ] [ ( 8) ] Eample 1:Simplify the epression: ( 1) Eample : Simplify the epression: 4 0 ( 1+ ) Try These 1: Simplify the epression: ( 1) 1 0 ( 4+ ) Try These : Simplify the epression: ( ) 4 1 ( ) 006 Colorado State University Mathematics Department

4 ELM Tutorial Unit.4 Solving Literal Equations It is useful in applications to be able to change the roles of the independent and dependent variables in equations. Solve the following equation for : = y = y = y + = ( y + ) = y + = y + Eample 1: Solve the following equation for m: 1 n m =. 9 Eample : Solve the following equation for y: = + y Try These 1: Solve the following equation for y : =. y Try These : Solve the following equation for s: = t s 006 Colorado State University Mathematics Department 4

5 ELM Tutorial Unit. Multiply Binomial Epressions Note : For any real numbers a, b, and c: a ( b + c) = ab + ac. For any real numbers a, b, c, and d: ( a + b)( c + d) = ac + ad + bc + bd. Use the distributive property to epand the epression: ( + 1)( 4) ( + 1)( 4) ( ) + ( 4) + ( 1 ) + ( 1 4) ( a + b) = ( a + b)( a + b) = a + ab + b Eample 1: Use the distributive property to epand the epression: ( + )( + ). Eample : Use the distributive property to epand the epression: ( )( ). Try These 1: Use the distributive property to epand the epression: ( + 7)( ). Try These : Use the distributive property to epand the epression: ( 4)( + 4). 006 Colorado State University Mathematics Department

6 ELM Tutorial Unit.6 Factor Quadratics Equations Writing a polynomial as the product of two or more polynomials is called factoring the polynomial. Factoring the quadratic + b + c 1. Find the pairs of numbers whose product is c.. Of the pairs in (1), find the pair whose sum is b. Factor the following quadratic epression: Pairs whose product is 6: (1,-6) (-1,6) (,-) (-,). Pair whose sum is 1: (,-) Therefore 6 = +. ( )( ) Eample 1: Factor the following quadratic epression: + 6 Eample : Factor the following quadratic epression: 10 4 Try These 1: Factor the following quadratic epression: + 6 Try These : Factor the following quadratic epression: Colorado State University Mathematics Department 6

7 ELM Tutorial Unit.7 Solve Linear Word Problems Many real world situations can be modeled with linear functions. To earn money for a new car, Julie is making and selling bracelets. Her weekly costs for advertising and phone calls are $0 and each bracelet costs $1.0 to produce. If Julie sells the bracelets for $4 each, how many bracelets will she have to sell each week to break even (start earning a profit)? Model Julie s epenses (E) and her revenue (R) (where is the number of bracelets sold each week): E = R = 4 Julie will break even when her epenses equal her revenue: = 4 0 =. = 1 Thus if Julie sells twelve bracelets each week, she will cover her epenses and begin to earn a profit. Eample 1: Cameron went to the bank to get change for a $0 bill in $ bills and $1 bills. He got bills in all. How many $ bills did he get? Eample : A standard volleyball court has a perimeter of 180 feet. The length is 0 feet longer than the width. What is its length? Try These 1: A police car traveling at 70 miles per hour is chasing a slow moving van. The van has a 10- mile head start and is traveling at 40 miles per hour. How long will it take the police car to catch the van? Try These : The measure of one supplementary angle is 4 more than twice the measure of the other. Find the measure of each angle. (Two angles are supplementary if the sum of their measures is equal to 180.) 006 Colorado State University Mathematics Department 7

8 ELM Tutorial Unit.8 Use Function Notation If an equation is a function, then the equation can be written in function notation: y = f (). The point (, y) can be written as (, f ()). The notation f () is read f at or the function value (y) at. Given f () = - + 6, find f (-). Find the function value when = -. f ( ) = ( ) + 6 = = 1 Thus f (-) = 1, which is the point (-, 1). Eample 1: Given f () = - +, find f (-). Eample : Given f () = - 1 and that f () = 9, find. Try These 1: Given f () = + - 1, find f (). Try These : Given f () = -4 + and that f () = -7, find. 006 Colorado State University Mathematics Department 8

9 ELM Tutorial Unit.9 Determine the Slope of a Line (Graphical Approach) The slope of a line describes the direction and steepness of the line. The slope of a line is referred to as m. A line with a positive slope rises from left to right. A line with a negative slope descends from left to right. The following figure describes lines with various slopes: Match the lines on the graph with their appropriate slope m = 4. m =. m = 1. Line b has a slope of ¼. It is the only line that rises from left to right thus it must have a positive slope. It is also not as steep as a line with slope 1.. Line a has a slope of. It decreases from left to right and is steeper than a line with slope 1.. Line c has a slope of /. It also decreases from left to right yet is less steep than a line with slope Colorado State University Mathematics Department 9

10 ELM Tutorial Unit Eample 1: Which line on the graph has a slope of? Eample : Estimate the slope of the line m. y y = y = m A) B) C) D) E) Colorado State University Mathematics Department 10

11 ELM Tutorial Unit Try These 1: Which line on the graph has a slope of 7? Try These : Estimate the slope of the line m. m y = y = y A) B) C) D) E) Colorado State University Mathematics Department 11

12 ELM Tutorial Unit.10 Graph Lines (by graphing intercepts) Another method for graphing a line is to graph the - and y-intercepts (where the line crosses the -ais and y-ais). The following line has a y-intercept at (0, -6) and an -intercept at (, 0). Notice that the y- intercept occurs when = 0 and the -intercept occurs when y = 0. The -intercept is also known as the horizontal intercept, and the y-intercept is also known as the vertical intercept. y = 6 Match the following equation to its graph: 1 y = 1 a. b. c. 1 Find the intercepts of y = 1 Vertical Intercept ( = 0): 1 y = (0) 1 y = 1 The line should cross the y-ais at (0, -1). Horizontal Intercept (y = 0): 1 0 = = = The line should cross the -ais at (, 0). Therefore the graph of the line is C. 006 Colorado State University Mathematics Department 1

13 ELM Tutorial Unit Eample 1: Which graph most accurately represents the linear function y = + 1? 4 A) B) C) D) E) 4 4 Eample : Which graph most accurately represents the linear function y = +? A) B) C) D) E) Try These 1: Which graph most accurately represents the linear function y = + 1? 4 A) B) C) D) E) 1 Try These : Which graph most accurately represents the linear function y = +? A) B) C) D) E) 006 Colorado State University Mathematics Department 1