### Create a StudySoup account

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

Already have a StudySoup account? Login here

# Precalculus (Phys & Math) MATH 1113

GC&SU

GPA 3.59

### View Full Document

## 36

## 0

## Popular in Course

## Popular in Mathematics (M)

This 50 page Class Notes was uploaded by Mrs. Kara Jacobs on Monday October 12, 2015. The Class Notes belongs to MATH 1113 at Georgia College & State University taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/221931/math-1113-georgia-college-state-university in Mathematics (M) at Georgia College & State University.

## Reviews for Precalculus (Phys & Math)

### 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: 10/12/15

MATH 1113 PreCalculus Section 62 notes The equation for the circle meaning centered at the origin is H4 4 Degrees PW 4 Radians 77 Degrees Q Radians 77 J2 pg 44 e Degree I Degrees I R d39 Radians 7 I a ms 7 7 a I I I I 1 Plt 4 g I Plt 2gt Degrees 7 Degrees Q Radians 77 quota e I 7 g Radians 77 h H 7 I 7 I quot I z I 7 a I 7 r I 7 r 1 4 4 P4 4 Degrees Degrees 360 Radians 7 I 5 Radians 7 21 r T I g quota a T I I quot I I quota 7 I I 39 I i I u quot 7 z I 39u e a I Plt 4 Degrees 7 I Radians 77 I e I I P4 4 a 4 Degrees Degrees Radians 7 I e Radians 7 P 4 Degrees 7 Degrees Radians 77 Radians P 342 l lfl gave you Dgg s fill in the m1ss1ng values Radians 7 2 On the calculator press the mode button and set it on DEGREE if not there already Notice the other choice it RADIAN 3 What does sin 20 7 Cos 20 7 4 On the calculator find sin 6 and cos 6 for a few of the above angles SWITCH BACK AND FORTH BETWEEN RADIANS AND DEGREES UNDER THE MODE BUTTION TO ENTER SOME ANGLES AS RADIANS AND SOME AS DEGREES Write out What they equal near the boxes or circles above Unit circle approach Above we discovered the following 39 quot Right angle approach m Notice that jz 1 2 2 P xy at a45 angle 2 39 2 therefore sin 45 andcos45 2 Noticethat 121 2 2 Pxy l ata30 angle 2 392 therefore sin 30 and cos 30 2 Noticethat 121 2 2 Pxy l ata60 angle 239 2 therefore sin 60 and cos 60 sin 0 ycoordinate of a point Pxy on the unit circle radius 1 cos 0 xcoordinate of a point Pxy on the unit circle radius l sinB X in this case r Likewise the following definitions are true for any circle of radius r is equal to the ycoordinate of a point Pxy on the unit circle cos 8 i in this case i is equal to the xcoordinate of a point Pxy on the unit circle r r The other 4 t 39 t 39 functions are defined as follows on the unit circle tang xii cot yii csc yii sec xii Likewise the following definitions are true for any circle of radius r tang xii cot yii csc yii sec xii Find sin 0 543 P25 5 2 X2 y21 5 Find cos 0 Find cos 0 Quadrantal angles 270 Angles of 30 45 60 7239 Find the exact values of the following after drawing the angle on the unit circle a cos 210 b sin7 c cscS d tan 7 4 6 4 Use the calculator to nd the following Draw the angle before calculating a sin 520 b tan 0 sec 6 Determine whether the following is positive or negative without using a calculator Then check your answer This is to help you understand the definition of a radian from section 61 a cos 3 b cos 1 0 sin 6 Give the area of the following triangle from the unit circle as a function of the angle 0 Find the area of the above triangle if H 30 by using the above formula M the formula Albh 2 Give the area of the following triangle NOT from the unit circle as a function of the angle 0 Find the area of the above triangle if H 30 by using the above formula m the formula Albh 2 MATH 1113 PreCalculus Section 42 notes A is in the form Rx Where p and q are functions and q is 7 the 7 poilsynomial The domain is the sqet of all real numbers except those for which the Example 1 domain Example 2 domain Example 3 domain Example 4 domain Example 5 domain Example 6 domain Horizontal and vertical asymptote introduction Analyze the graph of if i Now use transformations to graph the rational function 7 Vertical Asymptotes Rational functions have 77 7 7 77 at every place the function is 7 7 The graph 77 7 crosses a vertical asymptote l 1 Example 1 Example 2 2 3 x x 1 x2 4 Example 3 MATH 1113 PRECALCULUS SECTION 64 NOTES Graphs of the Sine and Cosine functions We will rst graph or Characteristics of the Sine Function 1 The domain is 2 The range is 3 The Sine function is an odd even or neither function Therefore it is symmetric with respect to the 4 The Sine function is periodic with period 5 The xintercepts are x y intercept is 6 The maximum value ofy is and occurs at x The minimum value of y is and occurs at x Graph functions of the form y Asina1x Example 1 Graph y3sinx l M y sin x 7N lt x T he Amplitude y Asinax y 3sin x A Amplitude This means 73g3sinxg3 Whereas ilgsinxgl for ysinx The Period of y Asinax The peIiod ofthe function y A SiII am is therefore for the above equationy 3 sinx w and the peIiod is Graph functions of the form y Asina1x Example 2 Graph y sin 4x Amplitude Period therefore the graph repeats every instead of To graphy sin x we start With 00 and notice a max a an xintercept at a min at 7 Back to an xintercept at All ofthe above are multiples ofthe period divided by 4 Use this fact to graph y sin 4x Example 3 Graph y 72 sinx Graph functions of the form y Asinwc Example 4 Find an equation for the graph The Graph ofy cosx Fill in the following table REPORTING BOTH DECIMAL AND EXACT VALUES in7m2 o I7r6I7r4I7r3I7r2I 7r I37r2I27r 5m Plot all the points to graph y cos x Put arrows at the ends of the graph to show that the graph continues Label every point above that is not a maximum pornt or an xintercept ycosx Characteristics of the Cosine Function 1 The domain is 2 The range is 3 The cosine function is an odd even or neither function Therefore it is symmetric With respect to the axis 4 The Cosine function is periodic With period 5 The xintercepts are x 4 yintercept is 6 The maximum value ofy is and occurs atx 27g The minimum value ofy is and occurs at x 3 MATH 1113 PreCalculus Section 61 notes If two lines are drawn with a common vertex they form an 7 One of the rays of an angle is called the and the other the Draw picture a Counterclockwise rotation Positive Angle Clockwise rotation Negative Angle An angle 6 theta is said to be in 7 if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive xaxis Draw picture a When an angle 6 is in standard position the terminal side either will lie in a quadrant in which case we say 6 7 7 7 7 or it will lie on the xaxis or the yaxis in which case we say 6 is a Draw aiamples a Angles are commonly measured in either or The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself 1 revolution is said to measure 360 degrees abbreviated A 7 is an angle of 7 revolution or 7 A 7 is an angle of 7 revolution or Draw the following angles 45 90 225 405 A degree breaks up into minutes and seconds the same way an hour breaks up into minutes and seconds i There are 7 7 minutes in degree ii There are 7 7 seconds in 77 minute iii There are 7 7seconds in 77 degree like an hour Therefore it follows from i that 7 7 7 same as 1 degree 60 minutes ii that 7 7 7 7 same as 1 minute 60 seconds iii that 7 7 7 7 same as 1 degree 3600 seconds 34 minutes is denoted as 28 seconds is denoted as 7 Examples 1 Convert 50 6 21 to a decimal in degrees 2 Convert 21256 to D M S form De nition ofa RADIAN Consider a circle of radius r Construct an angle whose vertex is at the center of this circle and whose rays create an arc on the circle whose length is r The measure of such an angle is 7 7 Draw picture a Arc Length For a circle of radius r a central angle of 6 radians subtends an arc whose length s is Example 1 Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 05 radians Draw a sketch for this example Example 2 Find the length of the arc of a circle of radius 10 miles subtended by a central angle of 2 radians Draw a sketch for this example Relationship between degrees and radians 1 revolution degrees i radians 12 revolution degrees i radians M revolution degrees i radians Since 180 TE radians it follows that Examples MATH 1113 PreCalculus Section 61 notes If two lines are drawn with a common vertex they form an 7 One of the rays of an angle is called the and the other the Draw picture a Counterclockwise rotation Positive Angle Clockwise rotation Negative Angle An angle 6 theta is said to be in 7 if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive xaxis Draw picture a When an angle 6 is in standard position the terminal side either will lie in a quadrant in which case we say 6 7 7 7 7 or it will lie on the xaxis or the yaxis in which case we say 6 is a Draw aiamples a Angles are commonly measured in either or The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself 1 revolution is said to measure 360 degrees abbreviated A 7 is an angle of 7 revolution or 7 A 7 is an angle of 7 revolution or Draw the following angles 45 90 225 405 A degree breaks up into minutes and seconds the same way an hour breaks up into minutes and seconds i There are 7 7 minutes in degree ii There are 7 7 seconds in 77 minute iii There are 7 7seconds in 77 degree like an hour Therefore it follows from i that 7 7 7 same as 1 degree 60 minutes ii that 7 7 7 7 same as 1 minute 60 seconds iii that 7 7 7 7 same as 1 degree 3600 seconds 34 minutes is denoted as 28 seconds is denoted as 7 Examples 1 Convert 50 6 21 to a decimal in degrees 2 Convert 21256 to D M S form De nition ofa RADIAN Consider a circle of radius r Construct an angle whose vertex is at the center of this circle and whose rays create an arc on the circle whose length is r The measure of such an angle is 7 7 Draw picture a Arc Length For a circle of radius r a central angle of 6 radians subtends an arc whose length s is Example 1 Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 05 radians Draw a sketch for this example Example 2 Find the length of the arc of a circle of radius 10 miles subtended by a central angle of 2 radians Draw a sketch for this example Relationship between degrees and radians 1 revolution degrees i radians 12 revolution degrees i radians M revolution degrees i radians Since 180 TE radians it follows that Examples MATH 1113 Precalculus Section 44 notes Solve x2 lt 4 by using the graph Steps for Solving Polynomial and Rational Inequalities 1 Write the inequality so that the polynomial or rational expression is on the left and zero is on the right 2 Determine when the expression on the left is equal to zero and with a rational expression is zero 3 Use the numbers found in step 2 to separate the real number line into intervals 4 Select a number in each interval and determine if the function on the left is positive gt 0 or negative lt 0 If the inequality isn t strict include the solutions to x 0 in the solution set Now solve the above equation using the procedures outlined above Steps 1 2 Zero s and unde ned points 4 Solution set in interval form EXAMPLES 1 Solve x 2 gt 0 2 Solve the inequality x3x 4 lt0 x x l I 2 2 321 4Solve gti x 3 Solve the inc uali q ty x 4 x 3 x1 MATH 1113 Precalculus 33 notes Examples of quadratic functions is a function of the form where a b and c are real numbers and 1 qt 0 Why The domain of a quadratic function consists of ilx73r 2 7 Z 1 f0 1x 1 0 for u 1u and u 3 ax for a lt 039 Graphs 0 a quadratic funcuon39 Properties ofthe Quadratic function fxax bxca 0 AXIS of Vertex is symmetry highest point The Vertex Axis of svmmetrv gt Venex is Axis of Parabola opens up 1fa 0 the vertex is a lowest point symmetry point a Opens up b Opens down a gt 0 a lt 0 Parabola opens down if a lt 039 the vertex is a point Also state the domain and range 1 fx 3x2 2x For the following quadratic functions nd x and yintercepts axis of symmetry vertex and finally graph the equation 2 fx 7x2 4x 3 fxxZ 74x3 4 fx2xz9x7 If you can t factor a quadratic equation then you will need the to nd the The formula is The xlntercepts of a Quadratic Function 1 If the discriminant l2 7 4110 gt 0 the graph of fx 0x2 bx C has two distinct xintercepts so it crosses the x axis in two places 2 If the discriminant 72 4110 0 the graph of fx ax2 bx C has one x intercept so it touches the x axis at its vertex 3 If the discriminant l2 7 4110 lt 0 the graph of fx 0x2 bx C has no xintercept so it does not cross or touch the x axis 5 fxx277x1 6 fccZ 76xll We havejust looked at quadratic functions in the form axz bx c Quadrch function can also be in the form where is the Vertex ofthe parabola Horn chapter 2 we know how to graph any function ofthe form fx ax7h Z k Vertex form horizontal shift Vertical shift Vertical stretch or compression Iflalgt1thenyaxz is thanyxz Iflallt1thenyaxz is thanyxz Example For the following quadratic function find x andyintercepts axis of symmetry Vertex and finally graph the equation Also state the domain and range fc7c4Z 72 Example nd the equation ofthe following parabola Example Graph 2x2 8x 1 by transformations by putting in in the form ax h2 k To do this we must Notes on completing the square Now we will continue with the example Graph 2x2 8x 1 by transformations MATH 1 1 13 Precalculus 21 notes A isa Ifx and y are two between in these and if a exists between x and y then we say that x corresponds to y or that y on x V Maps and Ordered Pairs as Relations Representatives Honda 1 NomiDakota 39 Alaska 7 Arizona 8 California 53 Colorado 7 Florida 25 North Dakota 1 Definitions LetX and Y be two nonempty sets of real numbers A from X into Y is a relation that associates with each element of X element of Y The setX is called the of the function For each elementx inX the corresponding element y in Y is called the image ofx The set of all images of the elements of the domain is called the of the function Damam FUNCTION PBISDH Phone number Gimm Cullean Phoebe EXAMPLE Determining Whether 1 Relation Represents a Function Calories Fa Wendy39s Slngle 410 Wendy39s an Bacon Classic 580 Burger King Whopper 5m Burner King Chicken Sandwich 750 a McDonald s Big Mac 600 McDonald39s McClllckan 430 39 EXAMPLE Determining Whether 3 Relation Represents a Funnion Carats Price EXAMPLE Determining Whether Relation Represents a Function Determine Whether each relation represents a inction If it is a inction state the domain and range 2737471i3i2i271 2 3 4i 1 37 27 27 1 2 3 4i 3 3i 3 2 1 TEXAMPLE Delrrnmiiug Wiioiini ti ReInliun Riprvyxnrs n Funuiun Determine ifthe equation y it 7 3 de nes y as a function ofx 2 Determine ifthe equation t 2y2 1 de nes y as a function ofx Function Notation y 6 7 is a symbol for the function 7 is the 7 Variable 7 is the 7 Variable 7 is the Value of the function at Domain Domain a fix x2 Domain Domain b Fixi gm x gtGX3 Range 3 d 3U C W 17 Example For the functionfde ned by 3x2 2x evaluate af3 bfx f3 cfx d fx e fx 3 D fxh fx h Finding the domain ofa function The of a function f is the set of real numbers such that the rule of the function makes sense Domain can also be thought of as the set of for the function Find the domain of each of the following functions a f x x 4 Example Express the area of a circle as a function 2 of its radius x 2x 3 b gx x2 9 The dependent variable is iand the independent variable is 7 c hx J3 2x The domain of the function is MATH 1113 PRECALCULUS SECTION 71 NOTES Precalculus review The following is background information needed in order to understand the inverse Sine Cosine and Tangent functions Functions A from X to Y is a relation that associates with each element of X ielement from Y Examples of functions 2 Stale popmagmn A l39m IliXDLLIJIILy D 11 lndlana 6159066 quot Cmquot Washington 50688 0 Duck 1 South Dakota 761063 7 Lion quot Nolh Ca I39na 8320146 I 0 Pi I Tennessee 5797289 Rabbit 1 7 Domain 3 y x Identity Function A quick way to see if an equation is function by de nition is to graph the function and determine if any single x value is associated with more than one y value Also know as the vertical line test Onetoone Functions A function IS if any 2 different inputs in the domain correspond to 2 different outputs in the range In other words a function IS NOT 7 if 2 different inputs correspond to the 7 output Which ofthe above functions 15 above is ll 7 Inverse functions A functlon has an i state Populalion Populallun Slate if and only if the function is Inmana 6159068 6159068 Indiana Washington 6068996 6068996 Washington South Dakota 761063 751063 gt SuumUakum The notatlon for an 1nverse functlon NorthCamlina 8320146 8320146 Norm Carolina is Tennessee 5797289 5797289 Tennessee Ii x x2 one to one Ii x x2 x 2 0 one to one If so nd the inverse Grth both functions Since f2 if i thenf 391L7 7 END REVIEW Graph K sin x 1 Is y sin x onetooneand therefore have an inverse 7 7 if 2 Find an interval that includes x 0 such that y sin x is onetoone Therefore y sin x on the interval i i is 11 and therefore has an 3 Find the inverse ofy sin x xysinx Domain Range x 9 fx ysinx f391x y s n39 x Domain Range x e f 400 1 ysm x Example 1 Find sin39ll 3 Example 2 Find sm391 J Graph both functions 1 Example 3 Find sm391 If functionsfand g are inverses then Examples 1 sin391sin 3 2 sin 391 sin 2 7 7777 3 3 sinsin39l 05 7 7777 4 sinsin 391 15 i 7777 Graph K cos x 1 Is y cos x onetooneand therefore have an inverse 7 7 7 2 Find an interval that includes x 0 X nonnegative such that y cos x is onetoone Therefore y cos x on the interval 7 i is 11 and therefore has an 3 Find the inverse ofy cos x 1 y COS x means 777 fxycosx Domain Range x 9 fx ycosx f391x y cos391 x Domain Range x e f 400 1 y cos x 5 Example 1 Find cosl7 2 Example 2 Find cos391 Graph both functions 2 Example 3 Can 005391 7239 Example 4 cos391 cos Example 5 coscos391 05 MATH 1113 Precalculus 51 notes Introduction Suppose that an oil tanker is leaking oil and we want to be able to determine the area of the circular oil patch around the ship It is determined that the oil is leaking from the tanker in such a way that the radius of the circular oil patch around the ship is increasing at a rate of 3 feet per minute Given two functions f and g the 7 denoted by i read as f composed with g is de ned by The domain off 0 g is the set of all numbers x in the domain of if such that if is in the domain of if Example 1 Given the functions f and g nd f 0 g2 Example 2 Given the functions f and g nd f 0 g and then nd the domain of this function Example 3 Find functionsfand g so thatfo g H MATH 1113 Objectives 39 Calculate and Interpret the Slope of a Line 39 Graph Lines Given a Point and the Slope 39 Use the PointSlope Form of a Line Find the Fauation of a Line Given Two Points Precalculus 13 Supplemental notes Write the Equation of a Line in SlopeIntercept Forrn De ne Parallel and Perpendicular Lines Find Equations ofParallel Lines Find Fauation ofPerpendicular Lines Let 7 7and 7 be two distinct points with 7 The 7 m of the nonvertical line L containing P and Q is de ned by the formula If x1 x2 L is a7 and the slope m ofL is 7 since this results in division by 0 Slope can be though of as the ratio ofthe change to the change 7 often termed Any two distinct points on a line can be used to compute the slope of the line or form Q to P because Example 1 Find and interpret the slope ofthe linejoining the points 14 and 23 1 When the slope ofthe line is 7 the line l l XAMPLE 7fromle toright Finding the Slopes of Various Linus CUIILJimIIg rhr Same Pnnn i2 3 Compute the slopes ofthe lines L 112113 andLA containing the 2 When the 510136 0mm line i5 77 the line renewing pairs ofpoints Graph all four lines on the same set of from le to right coordinate axes if i 39 Li P rz t Qi irlmli a It I P 7 P Q 7 H I a 7 if 3When the slope ofthelineis 7theline is L P Qi53 he new gt 7777 Li Pill grins H 0i 7t Hf 7quot 4 When the slope is7 7 the line is If x1 95271116 is 7 and me Slope is Example 2 Draw the graph ofthe equation x 2 7 Plotting the two points result in the graph of a vertical lin with the equation x x1 Qto39nyg HM MATH 1113 Precalculus 43 notes To analyze the graph of a rational function a Find the Domain of the rational function b Locate the intercepts if any of the graph c Test for Symmetry If Rx Rx there is symmetry with respect to the yaxis If Rx Rx there is symmetry with respect to the origin d Write R in lowest terms and nd the real zeros of the denominator which are the vertical asymptotes e Locate the horizontal or be able to state that there may be oblique asymptotes f Determine where the graph is above the x axis and where the graph is below the x axis g Use all found information to graph the function Example 1 Analyze the graph of Rx x l x2 4 Example 2 The concentration C of a certain drug in a patients bloodstream t minutes after injection is 50t ivenb Ct g y 0 t225 a Find the horizontal asymptote of Ct b What happens to the concentration of the drug as t time increases c Use a graphing utility to graph Ct According the the graph when is the concentration of the drug at a maximum MATH 1113 PRECALCULUS SECTION 63 NOTES Trigonometric De nitions unit circle sint9y c0st9x tant9 x 0 x csct9ly 0 sect9lx 0 c0tt9 y 0 y x y Tri onometric Functions sint97 c0st97 tant97 csct97 sect97 c0tt97 Domain of each of the tri functions What can 7 be Is there any place 7 is not defined draw unit circle 1 sin 9 y or R09 sint9 Domain 2 cos 9 x or R09 cos 9 Domain 3 tan 9 X or R09 tant9 Domain x 1 4 sec 9 0r R09 sect9 Domaln x 5 cot 9 i or R09 c0tt9 Domain y 1 6 csc 9 0r R09 csct9 Domaln y Range of each of the trig functions What can 7 be 1 sin 9 y or R09 sint9 Range 2 cos 9 x or R09 c0st9 Range 3 csc 9 i or R09 csct9 Range y 4 sec 9 l or R09 sect9 Range x 5 tan 9 Z or R09 tant9 Range x 6 cot 9 i or R09 c0tt9 Range y A function f is called if there is a positive number p such that whenever 6 is in the domain off so is and If there is a smallest such number p this smallest value is called the off Periodic properties sint927r sint9 cost927r cost9 tan tant9 csct927r csct9 sect927r sect9 cot cott9 Examples Find the exact value of the following by using the above properties Make a list in each quadrant indicating which if the trig function will be positive or negative y Reciprocal Identities csct9 sect9 cott9 Quotient Identities tan 9 cot t9 Example Given sin 9 and cos 9 find the value of each of the four remaining trigonometric functions of 9 Other useful identities The equation 01 thc unit circle is x y 1 But y sin 9 andx cos 6 so Example Find the exact value of each expression Do not use a calculator 1 cos 2 a cos 35 b foot 0502350 7 3 Sln g Example Given that 111 H 3 and cos H lt ll 11nd the exact values of each of the remaining five trigonnmcn ic functions MATH 1113 If I u I p A to FY quot PreCalculus Section 53 notes Suppose that are dreaming and I offer to pay you each day for attending class Being even more generous in quot I offer you a choice Choice 1 7 Receive 1000 per class attended Choice 2 7 Receive 2 cents on the first day you attend class 4 cents the second day 8 cents the third day and so on Before we go on which choice do you choose 1 Determine a formula for choice 1 2 What kind of equation did you create 7 3 How much money would you get for attending a 10 classes 777 7777 b 20 classes 77 777 c 30 classes 7 7777 4 Draw a graph of the money you make over 30 classes y 1 Let s determine a formula for the amount of money day you would receive on the x h 2 The equation 7 7 is an function 3 How much money would you get for attending a 10 classes 77 777 a 20 classes 77 77 b 30 classes 77 77 4 Draw a graph of the money you make over 30 classes y Review from algebra class Laws ofExponents lfs t a andb are real numbers with a gt 0 andb gt 0 en 1 41 2 7777 2 2713 77777 3 x3y61 3 7 For more review of these rules see the appendix A pages ASSA87 Exponential Functions The exponential function f with base a is 7 7 where a gt 0 and a 7i 1 Consider fx 3 gx 7 hx and rx Observations about graphs of exponential functions of the form f x a 1 A11 graphs pass through the point 77777 which is the 77intercept 2 77 7 is a horizontal asymptote 3 For bases gt 1 or for a gt 1 the functions are 77777 increasingdecreasing 4 For bases between 0 and 1 or 0 lt a lt 1 the functions are 7777 77 increasingdecreasing 5 For bases gt 1 the 777771argersma11er the base the steeper the graph 6 For bases between 0 and 1 the 777771argersma11er the base the steeper the graph 7 A11 graphs have domain 7 77 8 A11 graphs have range 77777 9 A11 graphs have 777 xintercepts 10 A11 graphs pass through the points and Using sketching techniques graph the following 1 fx 2 2 fx 2 2 3 fx 2M5 Domain Domain Domain Range Range Range HA HA HA 4 fx2quot 5 fx 2x 6 fx2quot 2 Domain Domain Domain Range Range Range HA HA HA The Base 2 The number 6 is defined as the number that the expression 1 x l x approaches as x a 00 In calculus thls IS expressed usmg 11m1t notatlon as e llm l xgtoo x Let s use the calculator to evaluate the above expression with the following values in the table a 11 x 100 1000 10000 100000 1000000 1000000000 e 2718281828459045235360287471352662497757247093699959575 N 6 N but not equal to MATH 1113 PRECALCULUS SECTION 64 NOTES Graphs of the Sine and Cosine functions We will rst graph or Characteristics of the Sine Function 1 The domain is 2 The range is 3 The Sine function is an odd even or neither function Therefore it is symmetric with respect to the 4 The Sine function is periodic with period 5 The xintercepts are x y intercept is 6 The maximum value ofy is and occurs at x The minimum value of y is and occurs at x Graph functions of the form y Asina1x Example 1 Graph y3sinx l M y sin x 7N lt x T he Amplitude y Asinax y 3sin x A Amplitude This means 73g3sinxg3 Whereas ilgsinxgl for ysinx The Period of y Asinax The peIiod ofthe function y A SiII am is therefore for the above equationy 3 sinx w and the peIiod is Graph functions of the form y Asina1x Example 2 Graph y sin 4x Amplitude Period therefore the graph repeats every instead of To graphy sin x we start With 00 and notice a max a an xintercept at a min at 7 Back to an xintercept at All ofthe above are multiples ofthe period divided by 4 Use this fact to graph y sin 4x Example 3 Graph y 72 sinx Graph functions of the form y Asinwc Example 4 Find an equation for the graph The Graph ofy cosx Fill in the following table REPORTING BOTH DECIMAL AND EXACT VALUES in7m2 o I7r6I7r4I7r3I7r2I 7r I37r2I27r 5m Plot all the points to graph y cos x Put arrows at the ends of the graph to show that the graph continues Label every point above that is not a maximum pornt or an xintercept ycosx Characteristics of the Cosine Function 1 The domain is 2 The range is 3 The cosine function is an odd even or neither function Therefore it is symmetric With respect to the axis 4 The Cosine function is periodic With period 5 The xintercepts are x 4 yintercept is 6 The maximum value ofy is and occurs atx 27g The minimum value ofy is and occurs at x 3 MATH 1113 Precalculus 23 Supplemental notes I Ob ectives Determine Even and Odd Functions from a Graph Identify Even and Odd Functions from the Equation Determine Where a Function is Increasing Decreasing or is Constant Locate Maxima and Minima Find the Average Rate of Change of a Function Even function A function f is if for every number 7 in its domain the number is also in the domain and Or in other words a function is if and only if whenever the point is on the graph of f then the point is also on the graph A function is if and only if its graph is with respect to the Odd function A function f is odd if for every number x in its domain the number x is also in the domain and So for an function for every point on the graph the point is also on the graph A function is if and only if its graph is with respect to the 39EXAMPLE Determining Even and Odd Fumtions from the Graph Determine whether each graph given is an even function an o d function or a function that is neither even nor odd EXAMPLE Identifying Even and Odd Functions Algebraically Determine whether each of the following functions is even odd or neitherThen Llc ermine whether the graph is symmetric with respect to the ynxis or with respect to the origin a f I Y 7 5 b gx 395 C I1 5 x 39 d th xi EXAMPLE Determining Where a Function ls Inm easing Decreasing or Constant from Its Graph A function f is 7 increasing or decreasing on an open I if for any choice of and in I with we have A function f is 7 increasing or decreasing on an open interval 1 if for any choice of and 7 in I with 7 we have A function is on an open interval if for all choices of x in the values fx are o 6 y p A function f has a if 7J1quot there is an interval I containing 0 so that for all x in I 0 3970 f0 e if We call fc a local l I off o x P I increasing demeasing The local maxunum is fc and occurs at X y A function f has a if 7 if fc 1m rm 1 1 E X h h I decreasing increasmg The local minimum is fc and occurs atx E 1 E g 41 NB there is an interval I containing 0 so that for all x in I 7 We call c a local off EXAMPLE Finding Local Maxima and Local Minima from the Graph of a Function and Determining Where the Function ls Increasing Decreasing or Constant Find all local maximums and minimums of the function 8 7 3 2 6 l 5 4 0 1 2 4 6 8 3 72 2 1 73 0 4 4 s 8 75 76 72 3 r7 78 Local minimum at 7 7 Local maXimum at 7 The local minimum is The local maXimum is If 0 is in the domain of a function y fx the7 of f between 0 and x is de ned as In Calculus this expression is called the 77 7 if The 7 7 of a function can be thought of as the average 7 7 of the function the change is y rise over the change in X run y fx Secant Line L r EXAMPLE Finding the Average Rate of Change Find the uverugc rule ol39 chungc of f39 2 3x3 a From I 103 b From I to 5 c From 1 l0 7 MATH 1113 Precalculus 24 Supplemental notes I Objectives Graph the Functions in the Library of Functions Graph Piecewisedefined Functions The following library of functions will be used throughout the text Be able to recognize the shape of each graph and associate that shape with the given function Linear Functions The equation if 7 represents any nonvertical line with slope if and yintercept All vertical lines are represented as 7 if where k is some constant The Identity Function Constant Function If m 1 and b 0 then we have a diagonal line y x since In 77 Domain Domain Range Range xintercepts xintercepts yintercepts yintercepts Even or Odd Even or Odd The Square Function The Cube Function y Domain Domain Range Range xintercepts xintercepts yintercepts yintercepts Even or Odd Even or Odd Domain Domain Range Range xintercepts xintercepts yintercepts yintercepts Even or Odd Even or Odd Piecewise de ned functions When functions are de ned by more than one equation they are called Dom ain Range xintercepts yintercepts Even or Odd Example 1 The function f is de ned as x3 2Sxltl fx 3 xl x3 xgtl b Determine the domain off Example 2 Graph the following function f x i Dom ain Range xintercepts yintercepts Even or Odd a Find f1 f1 114 d Find the range of f from the graph found in part c lxz 4 ifxgtl ifol 24 Assignment l28ALL 283 SEVEN 4144ALL

### 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

#### "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."

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### 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.