### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# COLLEGE ALGEBRA (QL)(SSS) MATH 1050

Utah State University

GPA 3.66

### View Full Document

## 19

## 0

## Popular in Course

## Popular in Mathematics (M)

This 56 page Class Notes was uploaded by Benton Yundt on Wednesday October 28, 2015. The Class Notes belongs to MATH 1050 at Utah State University taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/230419/math-1050-utah-state-university in Mathematics (M) at Utah State University.

## Similar to MATH 1050 at Utah State University

## Reviews for COLLEGE ALGEBRA (QL)(SSS)

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/28/15

Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 ltlt Real Numbers gtgt Note The set of numbers used for counting is called the set of natural numbers 1 2 3 4 Note The next set of numbers most of us learn about is the set of whole numbers 01 2 3 4 Note The whole numbers do not allow us to describe everyday situations For example if the balance in your checking account is 30 and you write a check for 35 your checking account is overdrawn by 5 We can write this as 5 read negative five Note The set consisting of the natural numbers 0 and the negatives of the natural numbers is called the set of integers Note One way of visualizing the different sets of numbers is to use what is know as a number line Example Whole Numbers Natural Nu mbers Integ ers Note If two integers are added subtracted or multiplied together the result is always another integer This however is not always the case with division For example 5 divided by 10 is To permit divisions such as we enlarge the set of integers calling the new collection the set of rational numbers Note The set of rational numbers consists of all the numbers that can be expressed as a quotient of two integers with the denominator not 0 Note The number line also contains points that cannot be expressed as quotients of integers These numbers are called irrational numbers because they cannot be represented by rational numbers Examples of irrational numbers 7r z 31415 6 z 27182 J2 z 14142 Page 1 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 Defn The set of real numbers is the set of all numbers each of which corresponds to a point on a number line Defn A symbol usually a letter such as x y or z used to represent any unknown number is called a variable Note One important property of real numbers is that they are ordered We use the inequality symbols to describe the ordered relationship between real numbers Interval Notation Inequality Graph ab 4 3 a b 4 dab anltb 4 l I P 517 altxsb lt gt law 4 moo xgta lt gt J7 x s b lt 39 gt oobgt x lt b lt ooaoo 00 lt x lt oo 4 Example Write each inequality in interval notation and graph a 2Sxlt7 b xlt10 Example Use inequality notation to describe w is at most 3 Page 2 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 Defn An is a combination of variables letters and constants real numbers combined using the operations of addition subtraction multiplication division and exponentiation Defn The of an algebraic expression are those pans that are separated by addition Example of algebraic expression x3 6x2 5 Note If we give a specific value to a variable we can evaluate an algebraic expression To evaluate an algebraic expression means to find its numerical value once we know the values of the variables 2x y2 4y Example If x23 and y2 find 3x 1 ltlt Exponents gtgt Note Repeated multiplication can be written in Example Properties of Exponents Property man amn 2 an am a 3 a7 a 4 a0 1 5 abm ambm 6 amn amrt 7 Z mz M w Page 3 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Example evaluate each a 24 c 24 Example Simplify each of the following a m6 4 t3ss 3 c 5 1 2 1 e 8 Fall 2007 b 24 d 23 d 8 Both very large and very small numbers frequently occur in many fields of study For example each day 26000000 pounds of dust from the atmosphere settles on Earth the length of the AIDS virus is 000011 millimeters the distance between the sun and the planet Pluto is approximately 5906000000 kilometers and the mass of a proton is approximately 000000000000000000000000165 gram It can be tedious to write these numbers in this standard decimal notation so scienti c notation is used as a convenient shorthand for expressing very large and very small numbers Defn A positive number is written in scientific notation when it is expressed in the form a x10 where a is a number greater than or equal to 1 and less than 10 1 S a lt10 and n is an integer Page 4 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 Example Write in standard form without exponents a 87 x105 b 328 x10 6 870000 000000328 Example Write in scientific notation a 00571 b 2140000000 571x10 2 214x109 ltlt Polmomials and Factoring gtgt You are probably familiar with the unpleasant onset of a cold We catch cold when the cold virus enters our bodies where it multiplies Fortunately at a certain point the virus begins to die The algebraic expression 075x4 3x3 5 describes the billions of viral particles in our bodies after x days of invasion The expression enables mathematicians to determine the day on which there is a maximum number of viral particles and consequently the day we feel sickest The expression 075x4 3x3 5 is an example of a Defn A is a single term or the sum of two or more terms containing variables in the numerator with wholenumber exponents ie terms of the form anx where n is a whole number Example a 7x64x3 9x2 13x 5 b 9x59 5 6 Example Simplify 7x3 8x2 9x 6 2x3 6x2 3x 9 Reminderz You addsubtract polynomials by collecting like terms terms having the same variables to the same powers In the 1980s a rising trend in global surface temperature was observed and the term global warming was coined Scientists are more convinced than ever that burning coal oil and gas results in a buildup of gases and particles that trap heat and raise the planet s temperature The average increase in global surface temperature in degrees Celsius x years after 1980 can be modeled by the polynomial 1 21 2 127 1293 21 3 127 2 1293 7x 7x 7x7 x x 7x 10000 500 100 5 5000000 1000000 50000 This model predicts global warming will increase through the year 2040 Furthermore global warming will increase at the greater rates near the middle of the twenty first century Page 5 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 Example Multiply a 4y x2y 3x b 5x 62y 3 1 2 E 3 2 c gx 55x 10x 20 Note Reversibility of thought is found throughout algebra For example we can multiply polynomials and show that We can also reverse this process and express the resulting polynomials as Defn a polynomial containing the sum of monomials means finding an equivalent expression that is a product Note We use the distributive property to multiply a monomial and a polynomial of two or more terms When we factor we reverse this process expressing the polynomial as a product Example Factor a 27x2y3 9xy2 3xy 2 20 12 b i 2 37 4 5 7 3 gt15Cy 45Cy 35Cy c x2x 3 25x 3 Page 6 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 Example Factor by grouping a 2mn 8n3m 12 b 6y2 20w15yw 8y Note To factor x2 bx c 1 Find a pair of factors that have as their and as their a If c is its factors will have the sign as b b If c is one factor will be and the other will be Select the factors such that the factor with the larger absolute value has the same sign as b 2 Check by multiplying Example Factor each of the following trinomials a x25x6 b x2 5x6 c x25x 6 d xz Sx 6 e x2 7x8 Page 7 of 8 Refresher Course Math 1050 and 1060 Notes Set 1 Fall 2007 Example Factor each of the following trinomjals a 6x4 21x3 9x2 b 6y2 19y10 Note Difference of Cubes x3 y3 x yx2 xy yz Note Sum of Cubes x3 y3 x yx2 xy yz Example Factor each of the following a p24p4 b 4x2 9 c 2y3 16x3 Page 8 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 ltlt Line Equations in Two Variables gtgt Example The graph shows the estimated number of milligrams of a pain medication M in the bloodstream t hours after a 1000 milligram dose of the drug has been given a Estimate the amount of medication after 4 hours 5 A 800 3973 E 15 g 600 E 400 b After how many hours was the 39 i i i m G med1c1ne concentration the 200 highest c After how many hours was the concentration 600 milligrams Defn A linear equation in two variables is an equation that can be written in the form Ax By C where A B and C are real numbers and A and B are not both 0 Note A solution of a linear equation in two variables requires two numbers one for each variable The solutions are usually written as an ordered pair of the form my 1 Example ls 2 5 a solution to 5x 2y 20 7 w The rectangular coordinate system allows us to visualize relationships between two variables by connecting any equation in two variables with a geometric figure To plot ab Start at origin Move a units to the left if alt0 or right if agt0 From there move b units up if bgt0 or down if blt0 Page 1 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Note In mathematics is a real number that measures the steepness of a line Defn Slope is the ratio of the vertical change in y the m to the horizontal change in x the m Example The following graph shows the cost y in cents of an in state long distance telephone call in Massachusetts where x is the length of the call in minutes Find the slope of the line What does the slope represent y A 100 Cost of Call in cents Length of Call in minutes Note Slope can be interpreted as a rate of change It tells us how fast y is changing with respect to x Note The slopeintercept form of a nonvertical line with slope m and y intercept b is Example Find an equation of a line with slope 1 and y intercept 08 Example Find the slope of the line 2x 3y 12 Note The pointslope form of a nonvertical line with slope m that passes through the point x1 y1 is Page 2 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Example Find the equation of the line through the points 25 and 16 Write the equation in slope intercept form Example Find the equation of the line containing the point 35 perpendicular to the line 3x 2 y 10 Graph the line and determine algebraically whether the point 45 is on the line Page 3 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Example A New York city taxi service charges an initial fee of 200 and then 020 for every 15 mile traveled Determine the function representing the cost fare for a taxi ride of x miles and use it to find the cost of a 34 mile taxi ride ltlt Functions gtgt Defn A f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B The set A is the domain or set of inputs of the function f and the set B contains the range or set of outputs Example State whether the following describes a function Page 4 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Note 0 Each element of A must be matched with an element of B 0 Some elements of B may not be matched with any element of A 0 Two or more elements of A may be matched with the same element of B 0 An element of A the domain cannot be matched with two different elements of B Example The relationship between price and items in a supermarket is a function Note In algebra it is common to represent functions by equations or formulas involving two variables The equation y x2 represents the variable y as a function of the variable x In this equation x is the independent variable and y is the dependent variable The domain of the function is the set of all values taken on by the independent variable x and the range of the function is the set of all values taken on by the dependent variable y Example Does the equation y 3x 5 represent a function Example Does the equation y2 7x 8 represent a function Note When an equation is used to represent a function it is convenient to name the function so that it can be referenced easily We usually let y f x Example Let f x x2 2 Find each of the following 3 f 3 b f W C fxhZ fx Page 5 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Example Let fxix25 7 34 Find a f 5 b f3 C f4 Note The domain of a function can be described explicitly or it can be implied by the expression used to define the function Defn The implied domain is the set of all real numbers for which the expression is defined x is the set of all real numbers x 2 3 x Example The implied domain of f x Example Find the domain of f x v9 x2 ltlt Graphs of Functions gtgt Defn The graph of a function f in the xy plane consists of the points xy such that x is in the domain of f and y f x 3 Note A curve in the xy plane is the graph of a function if no vertical line intersects the curve more than once Page 6 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Defn A function f is increasing decreasing on an interval ab if for any two numbers x1 and xzin ab fx1 lt fx2 fx1 gt fx2 whenever x1ltx2 A function f is constant on an interval ab if for any x1 and x2 in ab fX1fxg Defn A function f has a relative maximum relative minimum at xc if there exists an open intervalab containing 0 such that f x S f c f x 2 f c for all x in ab Example Use the graph of the function f given below to answer the following questions y a domain A 4 b range 3 C f 2 d f 0 L I e fx4 whenxz m x N H H N w 4 K V 2 f decreasing g increasing h relative max i relative min Page 7 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Defn A function f is even if for each x in the domain of f f x f x Defn A function f is M if for each x in the domain of f f x f x Defn A graph has symmetry with respect to the yaxis if whenever xy is on the graph so is the point xy Defn A graph has symmetry with respect to the orig39n if whenever xy is on the graph so is the point x y Defn A graph has symmetry with respect to the xaxis if whenever xy is on the graph so is the point x y Note Example Determine whether the function is even odd or neither a fxx3 x b fxx21 c fxx3 1 Page 8 of 8 Refresher Course Math 1050 and 1060 Notes Set 6 ltlt Exponential Functions gtgt Defn Let a denote an arbitrary positive constant other than 1 The exponential function with base a is defined by f x ax The domain of f is oooo the range of f is 000 and the graph of f is as follows Note Properties of Exponential Functions Let a and b be positive numbers Then 1 a 1 2 axayaxy 3 jary 4 axyaxy d 5 aw aw 6 3 X 5 7 all b V ax Example Find f 4 if f x 2 1 Example Graph f x 1X Example Graph f x 5 2H Page 1 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Defn The natural exponential function is the exponential function with base 6 f x ex Note f x ex is often referred to as the exponential function Reminder e z 27182 Example Graph f x ex Example Graph f x eH 2 Example The number of bacteria in a culture is given by the formula nt 500604 where t is measured in hours What is the initial population of the culture ltlt Logarithmic Functions gtgt Note Exponential functions are one to one pass horizontal line test So they have inverses These inverses are known as logarithmic functions Defn Let b 2 1 b gt 0 Then 10gb x is the exponent to which I must be raised to yield x That is ylogbxltgtby x The domain is 000 the range is oooo and its graph is Page 2 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Example Evaluate log5 25 Example Evaluate log2 Example Graph log2 x Defn lnx means loge x which is the exponent to which 6 must be raised to yield x logx means log10 x which is the exponent to which 10 must be raised to yield x Example Evaluate lne and lnl Page 3 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 ltlt Properties of Logarithms gtgt Laws of Logarithms Let b be a positive number with b 1 Let m and n be positive numbers Then 1 10gb mn 2 10gb 3 10gb m 4 10gb1 5 log I 6 10gb bx xx2 m x2 1 Example Expand log2 Example Write 1n5 Zlnx 31nx2 5 as a single log with coefficient 1 Page 4 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Most calculators have only two types of log keys one for common logarithms base 10 and one for natural logarithms base 6 Although common logs and natural logs are the most frequently used you may occasionally need to evaluate logarithms to other bases Note Change of Base Formula Let a b and x be positive real numbers such that a 1 and b 2 1 Then log x can be converted to a different base using one of the following formulas 10gb x log10 x lnx logax i logaxi loga x log a log10 a lna Example Rewrite log 4 25 in terms of natural logs ltlt Solving Exponential and Logarithmic Eguations gtgt Example Solve each equation a 32X 42 b 46 3 2 Page 5 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Example Continuation Solve each equation C 63 102 217 d eZX eX 60 e xzexxeX ex0 f log6xlog6x10 g log22x2 45 h 1nxlnx1ln12 Page 6 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Defn A quantity that experiences exponential growth gdecayl increases decreases according to the formula Qt Qoek where t time Qt 2 quantity at time t Q0 2 initial size of quantity k 2 relative rate of growth decay Exponential growth is indicated by k gt 0 and exponential decay by k lt 0 Example The population of the world in the year 1650 was 470 million and in the year 1999 was 5996 million Assuming that the population of the world grows exponentially find the equation for the population Pt in millions in the year t Page 7 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 ltlt Exponential Functions gtgt Defn Let a denote an arbitrary positive constant other than 1 The exponential function with base a is defined by f x ax The domain of f is oooo the range of f is 000 and the graph of f is as follows Note Properties of Exponential Functions Let a and b be positive numbers Then 1 a 1 2 axayaxy 3 jary 4 axyaxy d 5 aw aw 6 3 X 5 7 all b V ax Example Find f 4 if f x 2 1 Example Graph f x 1X Example Graph f x 5 2H Page 1 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 Defn The natural exponential function is the exponential function with base 6 f x ex Note f x ex is often referred to as the exponential function Reminder e z 27182 Example Graph f x ex Example Graph f x eH 2 Example The number of bacteria in a culture is given by the formula nt 500604 where t is measured in hours What is the initial population of the culture ltlt Logarithmic Functions gtgt Note Exponential functions are one to one pass horizontal line test So they have inverses These inverses are known as logarithmic functions Defn Let b 2 1 b gt 0 Then log 7 x is the exponent to which I must be raised to yield x That is ylogbxltgtby x The domain is 000 the range is oooo and its graph is Page 2 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 Example Evaluate log5 25 Example Evaluate log2 Example Graph log2 x Defn lnx means loge x which is the exponent to which 6 must be raised to yield x logx means log10 x which is the exponent to which 10 must be raised to yield x Example Evaluate lne and ln1 Page 3 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 ltlt Properties of Logarithms gtgt Laws of Logarithms Let b be a positive number with b 1 Let m and n be positive numbers Then 1 10gb mn 2 10gb 3 10gb m 4 logb1 5 log I 6 10gb bx xx2 m x2 1 Example Expand log2 Example Write 1n5 Zlnx 3lnx2 5 as a single log with coefficient 1 Page 4 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 Most calculators have only two types of log keys one for common logarithms base 10 and one for natural logarithms base 6 Although common logs and natural logs are the most frequently used you may occasionally need to evaluate logarithms to other bases Note Change of Base Formula Let a b and x be positive real numbers such that a 1 and b 2 1 Then log x can be converted to a different base using one of the following formulas 10gb x log10 x lnx logax i logaxi loga x log a log10 a lna Example Rewrite log 4 25 in terms of natural logs ltlt Solving Exponential and Logarithmic Eguations gtgt Example Solve each equation a 32X 42 b 46 3 2 Page 5 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 Example Continuation Solve each equation C 63 102 217 d eZX eX 60 e xzexxeX ex0 f log6xlog6x10 g log22x2 45 h 1nxlnx1ln12 Page 6 of 7 Refresher Course Math 1050 and 1060 Notes Set 6 Fall 2007 Defn A quantity that experiences exponential growth gdecayl increases decreases according to the formula Qt Qoek where t time Qt 2 quantity at time t Q0 2 initial size of quantity k 2 relative rate of growth decay Exponential growth is indicated by k gt 0 and exponential decay by k lt 0 Example The population of the world in the year 1650 was 470 million and in the year 1999 was 5996 million Assuming that the population of the world grows exponentially find the equation for the population Pt in millions in the year t Page 7 of 7 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 ltlt Line Equations in Two Variables gtgt Example The graph shows the estimated number of milligrams of a pain medication M in the bloodstream t hours after a 1000 milligram dose of the drug has been given a Estimate the amount of medication after 4 hours 1000 800 600 b After how many hours was the medicine concentration the highest 400 Pain Medication in milligrams 200 c After how many hours was the concentration 600 milligrams Defn A linear equation in two variables is an equation that can be written in the form Ax By C where A B and C are real numbers and A and B are not both 0 Note A solution of a linear equation in two variables requires two numbers one for each variable The solutions are usually written as an ordered pair of the form my 1 Example ls 2 5 a solution to 5x 2y 20 7 w The rectangular coordinate system allows us to visualize relationships between two variables by connecting any equation in two variables with a geometric figure To plot ab Start at origin Move a units to the left if alt0 or right if agt0 From there move b units up if bgt0 or down if blt0 Page 1 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Note In mathematics is a real number that measures the steepness of a line Defn Slope is the ratio of the vertical change in y the m to the horizontal change in x the m Example The following graph shows the cost y in cents of an in state long distance telephone call in Massachusetts where x is the length of the call in minutes Find the slope of the line What does the slope represent y A 100 Cost of Call in cents Length of Call in minutes Note Slope can be interpreted as a rate of change It tells us how fast y is changing with respect to x Note The slopeintercept form of a nonvertical line with slope m and y intercept b is Example Find an equation of a line with slope 1 and y intercept 08 Example Find the slope of the line 2x 3y 12 Note The pointslope form of a nonvertical line with slope m that passes through the point x1 y1 is Page 2 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Example Find the equation of the line through the points 25 and 16 Write the equation in slope intercept form Example Find the equation of the line containing the point 35 perpendicular to the line 3x 2 y 10 Graph the line and determine algebraically whether the point 45 is on the line Page 3 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Example A New York city taxi service charges an initial fee of 200 and then 020 for every 15 mile traveled Determine the function representing the cost fare for a taxi ride of x miles and use it to find the cost of a 34 mile taxi ride ltlt Functions gtgt Defn A f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B The set A is the domain or set of inputs of the function f and the set B contains the range or set of outputs Example State whether the following describes a function Page 4 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Note 0 Each element of A must be matched with an element of B 0 Some elements of B may not be matched with any element of A 0 Two or more elements of A may be matched with the same element of B 0 An element of A the domain cannot be matched with two different elements of B Example The relationship between price and items in a supermarket is a function Note In algebra it is common to represent functions by equations or formulas involving two variables The equation y x2 represents the variable y as a function of the variable x In this equation x is the independent variable and y is the dependent variable The domain of the function is the set of all values taken on by the independent variable x and the range of the function is the set of all values taken on by the dependent variable y Example Does the equation y 3x 5 represent a function Example Does the equation y2 7x 8 represent a function Note When an equation is used to represent a function it is convenient to name the function so that it can be referenced easily We usually let y f x Example Let f x x2 2 Find each of the following 3 f 3 19 f W C fxhZ fx Page 5 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Example Let fxix25 7 34 Find a f 5 19 f3 C f4 Note The domain of a function can be described explicitly or it can be implied by the expression used to define the function Defn The implied domain is the set of all real numbers for which the expression is defined x is the set of all real numbers x 2 3 x Example The implied domain of f x Example Find the domain of f x v9 x2 ltlt Graphs of Functions gtgt Defn The graph of a function f in the xy plane consists of the points xy such that x is in the domain of f and y f x 3 Note A curve in the xy plane is the graph of a function if no vertical line intersects the curve more than once Page 6 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Defn A function f is increasing decreasing on an interval ab if for any two numbers x1 and xzin ab fx1 lt fx2 fx1 gt fx2 whenever x1ltx2 A function f is constant on an interval ab if for any x1 and x2 in ab fX1fxg Defn A function f has a relative maximum relative minimum at xc if there exists an open intervalab containing 0 such that f x S f c f x 2 f c for all x in ab Example Use the graph of the function f given below to answer the following questions y a domain A 4 b range 3 6 f 2 r d f0 0 e fx 4 when x m w M a N w 4 K V 2 f decreasing g increasing h relative max i relative min Page 7 of 8 Refresher Course Math 1050 and 1060 Notes Set 3 Fall 2007 Defn A function f is m if for each x in the domain of f f x f x Defn A function f is M if for each x in the domain of f f x f x Defn A graph has symmetry with respect to the yaxis if whenever xy is on the graph so is the point xy Defn A graph has symmetry with respect to the orig39n if whenever xy is on the graph so is the point x y Defn A graph has symmetry with respect to the xaxis if whenever xy is on the graph so is the point x y Note Example Determine whether the function is even odd or neither a fxx3 x b fxx21 c fxx3 1 Page 8 of 8 Refresher Course Math 1050 and 1060 Notes Set 5 Fall 2007 ltlt uadratic Functions gtgt DLfn A Quadratic function is a function defined by an equation of the form fxwc2 bxc where a b and c are real numbers and a 2 0 It can also be written as fxax h2 k a 0 where h and k are real numbers Its graph is a parabola It has vertex h k Example Graph f x x2 6x 7 Page 1 of 5 Refresher Course Math 1050 and 1060 Notes Set 5 Fall 2007 ltlt Polmomial Functions of Higher Degree gtgt Defn A polynomial function of degree n is a finite sum of nonnegative integer powers of x px anx aHx H azxz alx do where the a are constants and n is a whole number Key Characteristics of Graphs of Polynomial Functions 0 The domain is the set of all real numbers 0 There are at most n x intercepts 0 The ends of the graph point in the same direction if n is even 0 The ends of the graph point in opposite directions if n is odd Defn A zero of a function f is a number a for which f a 0 Also we have that x a is a factor of f Note For a polynomial function a factor of x ak k gt 1 yields a repeated zero xza of multiplicity k 0 If k is odd the graph crosses the x axis at xza 0 If k is even the graph touches the x axis but does not cross the x axis at xza Example Describe the graph of f x xx 2x 2x2 1 Example Describe the graph of f x x2 x 2x 1x2 1 Page 2 of 5 Refresher Course Math 1050 and 1060 Nola Set 5 Fall 2007 Example Flnd a polynomral functlon of mrnrmal degree havlng zero Pig 3 3 Example Determrne a polynomral of mlnlmum degree havlng zero at 72 0 3 pamng through the pomt 16 ltlt Rational Functions and A mgtotes gtgt Px D ln rrr L M Q wrth gape 1 Note Thepllmaryexampleofamtlomlfunctlonl y whoxe graphluho nat rlght Notlce that the functlon u unde ned an 0 the tench ofthe graph approach the mam by hand became they lequlre carefully plotted pom However we can eanly ldentlfy key charactellxtlc of uch graph baxed only upon the numerator p x and the denomrnator rim 2 Pages ofS Refresher Course Math 1050 and 1060 Notes Set 5 Fall 2007 Defn The line xza is a vertical asymptote of the graph of f if f x gt 00 or f x gt oo as x a a from either the left or from the right Example Find the veitical asymptotes y f x x3 Example Find the veitical asymptotes of f x 2 9 x Defn The line yzb is a horizontal asmptote of the graph of f if f x gt b as x a 00 or x a w Example Find the horizontal asymptotes y 157 y fx Page 4 of 5 Refresher Course Math 1050 and 1060 Notes Set 5 Fall 2007 Example Find the horizontal asymptotes of 15x26x 1 a foo 7x24 3x3 x21 b fX 6x4 x3 xz x C fx x1 xz x Z Example Sketch the graph of f x 1 Page 5 of 5 Refresher Course Math 1050 and 1060 Notes Set 5 ltlt uadratic Functions gtgt DLfn A Quadratic function is a function defined by an equation of the form fxwc2 bxc where a b and c are real numbers and a 2 0 It can also be written as fxax h2 k a 0 where h and k are real numbers Its graph is a parabola It has vertex h k Example Graph f x x2 6x 7 Page 1 of 5 Refresher Course Math 1050 and 1060 Notes Set 5 ltlt Polmomial Functions of Higher Degree gtgt Defn A polynomial function of degree n is a finite sum of nonnegative integer powers of x px anx aHx H azxz alx do where the a are constants and n is a whole number Key Characteristics of Graphs of Polynomial Functions 0 The domain is the set of all real numbers 0 There are at most n x intercepts 0 The ends of the graph point in the same direction if n is even 0 The ends of the graph point in opposite directions if n is odd Defn A zero of a function f is a number a for which f a 0 Also we have that x a is a factor of f Note For a polynomial function a factor of x ak k gt 1 yields a repeated zero xza of multiplicity k 0 If k is odd the graph crosses the x axis at xza 0 If k is even the graph touches the x axis but does not cross the x axis at xza Example Describe the graph of f x xx 2x 2x2 1 Example Describe the graph of f x x2 x 2x 1x2 1 Page 2 of 5 Refresher Course Math 1050 and 1060 Notes Set 5 Example Flnd a polynomral functlon of mrnrmal degree havlng zero Pig 3 3 Example Determrne a polynomral of mlnlmum degree havlng zero x 72 0 3 pamng through the pomt 16 ltlt Rational Functions and A mgtotes gtgt Px D ln rrr L M Q wrth gape 1 Note Thepllmaryexampleofamtlomlfunctlonl y whoxe graphluho nat rlght Nome that the functlon u unde ned an 0 the tench ofthe graph approach the mam by hand became they lequlre carefully plotted pom However we can eanly ldentlfy key charactemtlc of uch graph baxed only upon the numerator p x and the denomrnator rim 2 Pages ofS Refresher Course Math 1050 and 1060 Notes Set 5 Defn The line xza is a vertical asymptote of the graph of f if f x gt 00 or f x gt oo as x a a from either the left or from the right Example Find the veitical asymptotes y f x x3 Example Find the veitical asymptotes of f x 2 9 x Defn The line yzb is a horizontal asmptote of the graph of f if f x gt b as x a 00 or x a w Example Find the horizontal asymptotes y 157 y f x Page 4 of 5 Refresher Course Math 1050 and 1060 Notes Set 5 Example Find the horizontal asymptotes of 15x26x 1 a foo 7x24 3x3 x21 b fX 6x4 x3 xz x C fx x1 xz x Z Example Sketch the graph of f x 1 Page 5 of 5 Refresher Course ltlt Inverse Trigonometric Functions gtgt Recall various familiar pairs of functions have an inverse relationship For example fx e and f 1x In x note the notation f 1 is used to denote the inverse off are inverses because 1116 x and 6quot x for all x in the domains of the respective functions Not coincidentally the graph of one is the re ection of the graph of the other through the main diagonal y x see figure Similarly fx x3 and f 1x 35 are inverses because 3 x and x for all x Again the graph of each is the re ection of the other through the main diagonal One is tempted to conclude f x x2 and gx J are inverses but such is not the case For C 2 x is not true for all xitry x 1 Furthermore the graph of one is not the re ection of the other ie the portion of the graph of f x x2 in the second quadrant does not re ect onto the graph of gx J Page 1 of8 Math 1050 and 1060 Notes Set 9 5 ltxgtZ Refresher Course Math 1050 and 1060 Notes Set 8 Note Continuation However when the domain of for x2 is restricted to the nonnegative real numbers 2 x x1 x 2 0 fx x2 x 2 O and f39lx J are inverses Certainly Jx Zxand slyzx foralleO 5 Furthermore the graph ofone now is the re ection ofthe other through the main diagonal Restricting fquotx J the domain in this way results in a function whose graph not only passes the vertical line test but the horizontal line test as well Such a function is called a onetoone function Since the graph of a onetoone function passes the horizontal line test it s re ection through the main diagonal will pass the vertical line test and therefore will be the graph of a function m The trigonometric functions are much like fx x2 in the sense that their domains must be restricted for them to be onetoone so they will have inverse functions There are many such ways to restrict the domains of the trigonometric functions but it is conventional to restrict their domains as follows fx sin 6 l f t r fx cosx 0 7 Page 2 of8 Refresher Course Math 1050 and 1060 Notes Set 8 Note Every onertorone function f including those pictured above has an inverse function f 1 such that fquot fc c andf fquotc c for all x in the domains of the respective functions causing the graph of fquot to be the re ection of the graph off through the main diagonal Note Therefore f x sin 1 has an inverse function called the arcsine function denoted arcsin c or sinquot 5 Since for sin 1 and the arcsine function are inverses We have arcsinsin c 5 quotso long as c is in the 5 77 7r restricted domain and sinarcsin c c for all x in the domain of the y amsin x arcsine function Furthermore the graph of the 1 arcsine function is the re ection of the graph offltxgtsinx 11 mroughthemain 2 2 7 d I diagonal T 35 7057 y arccosx 7 Inverse functions for the other trigonometric functions are similarly defined have similar names notations and properties and have graphs obtained in similar Ways 5 y arctanx Refresher Course Math 1050 and 1060 Notes Set 8 Defn The inverse sine function is defined by y arcsin x if and only if sin y x where 1 S x S1 and S y S The domain of y arcsinx is 11 and the ran eis E g 2 Example Find each of the following i 1 a arcsm b meant23 c arcsin4 Defn The inverse cosine function is defined by y arccos x if and only if cos y x where 1 S x S1 and 0 S y S 7 The domain of y arcsinx is 11 and the range is 0 Defn The inverse tangent function is defined by y arctanx if and only if tan y x where oo lt xlt co and lt y lt The domain of y arcsinx is oooo and the 0 7r 7r rangeis ii l 2 2 Page 4 of 8 Refresher Course Math 1050 and 1060 Notes Set 8 Note Summaiy of the graphs of the three inverse trigonometric functions described Example Find the exact value a arccosT2 b arctan 1 2 c tan arccosg Page 5 of 8 Refresher Course Math 1050 and 1060 Notes Set 8 ltlt Solving Trigonometric Eguations gtgt Example Find all solutions of sin x f sin x in the interval 027r Example Solve 3tan2 x l 0 Example Solve cotxcos2 x 2cotx Page 6 of 8 Refresher Course Math 1050 and 1060 Notes Set 8 Example Solve 2sin2 x3cosx 3 0 Example Find all solutions of cos x 1 sin x in the interval 027r Page 7 of 8 Refresher Course Math 1050 and 1060 Notes Set 8 Example Find all solutions of 2cos x sin 2x 0 Example Find all solutions of sin x sin x 1 in the interval 027 Page 8 of 8

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.