Name Date Period Sections and Scoring Pre-Calculus Midterm Review Packet (Chapters,, ) Your midterm eam will test your knowledge of the topics we have studied in the first half of the school year There will be two sections a calculator free section and a graphing calculator section The eam consists of a variety of multiple choice and open-ended questions There is no formula sheet, so please make sure you know any necessary formulas for the eam (Growth, decay, and compound interest formulas will be given) How do I prepare for the Midterm Eam? Answer as many problems as possible from this review packet and check your answers You should only use a calculator to answer the questions marked with the calculator symbol: If you do not own a graphing calculator, use the link on my website or use desmoscom to practice Do not wait until the last minute to begin reviewing for this eam Review old tests and quizzes from the st and nd marking periods IF YOU TAKE THE TIME TO PREPARE FOR THIS EXAM, YOU MAY BE PLEASANTLY SURPRISED WITH HOW MANY QUESTIONS YOU CAN ANSWER ON THE MIDTERM Eam Schedule Date Session Session Tuesday, January 9th Wednesday, January 0th Thursday, January st Friday, February st 4/5 5/6 6/7 8/9 7/8 9/0 CHAPTER REVIEW PROBLEMS (All Sections) Find each functional value for: ) f ( ) ) f ( ) State the domain of each function using interval notation 9 5 ) 4) f ( ) ( ) [ ],,, if if if < < 4
Find each functional value if c ( ) 8 and m( ) 5 6 5) c( d ) 6) m ( ) m 8) c ( ) 7) ( ) Find the y-intercept and the zeros of each function using algebraic methods 9) g ( ) 5 0) h( ) Use the graph of given below to answer questions -5 ) State the domain using interval notation ) State the range using interval notation ) f ( ) lim lim What type of discontinuity is there at? 4) f ( ) lim lim What type of discontinuity is there at? 5) Describe the end behavior using limit notation 6) Determine if h( ) 5 is even, odd, or neither Confirm algebraically f f (symmetric with y-ais) f f (symmetric with origin) To be an even function, for every in the domain of f, ( ) ( ) To be an odd function, for every in the domain of f, ( ) ( ) 7) Use the Continuity Test to determine if the function is continuous at the given value of If the function is discontinuous at the given value of, identify the type of discontinuity as infinite, jump, or removable ; at 4 f() A table of values is shown for a cubic function over the interval [, 4] 8) Determine the y-intercept 9) Determine between which consecutive integers the real zeros are located y - -4 - -5 0 - - 4 90
Complete each of the following for the given function, f () 0) Name and classify the etrema ) Describe the end behavior using limit notation ) State the interval(s) for which the function is increasing ) State the interval(s) for which the function is decreasing 4) State the interval(s) for which the function is constant 5) State the domain using interval notation 6) State the range using interval notation Graphing Calculator Skills Practice 4 7 Recommended Viewing Window: [ 5, 5, ] by [ 6,0, ] 7) Approimate the zero(s) to the nearest hundredth 8) Name and classify the etrema Approimate to the nearest hundredth d t 7t, 9) The formula for distance traveled by falling objects on the Moon is ( ) where d ( t) is the distance in feet and t is the time in seconds Find the average rate of change (average speed) of the object over the time interval [, ] Transformations: Know how to make all si parent functions and how to perform transformations on them! [ ] [ ] A parent function is given Graph the new function without making a table of values 0) Parent: ) Parent: [ ] ) Parent: New: ( ) ( ) g 5 New: ( ) [ ] New: g ( ) ( ) g
) Parent: 4) Parent: y 5) Parent: y New: g ( ) New: y f ( ) New: y f ( ) 6) If is compressed vertically by a factor of 5, reflected over the y-ais, and translated 7 units down, write an equation for the new function g () 7) What transformations must go through to become g ( ) 4 8? Use and ( ) 6 g to answer each of the following questions g 8) ( f g)( ) 9) ( g f )( ) 40) ( f g)( ) 4) ( f )( ) g 4) [ go f ]( ) 4) ( g f )( 5) 44) [ go f ]( 5) 45) domain of ( )( ) f Use the graph at the right to answer questions 46-49 46) Write the equation for g () g( ) 47) Is this an even function, odd function, or neither? 48) Apply the Horizontal Line Test to determine whether its inverse function eists Write yes or no 49) Draw the graph of y g() Find [ g]( ) f o and the domain of f o g 50) and g( ) 5) and g( ) 5 f 5) Find an equation for, the inverse o 4
Answer the following questions using the function ( ) f 5) State the domain and range of using interval notation 54) Find the equation for 55) State the domain and range of using interval notation The current average echange rate from Euros to US dollars can be described by 09, where is the currency value in Euros and is the currency value in US dollars 56) Eplain why 57) What do and eists represent in the inverse function Study Tip Break the study guide problems up in to sections, and do a section each day If you have any questions, come to etra help 58) Graph the inverse of the given function 59) Graph the given piecewise function State the domain and range h ( ) 5,,, if < if < 0 if 0 CHAPTER ANSWERS ) 0 ) ( 5 5 ) (, 7) ( 7, ) 4) [, 4) ( 4, 0 5) 9d 6d 8 6) 5 7) 0 8) 4 8 5 9) y-intercept: 4 0) y-intercept: 0 zeros: 4, 6 zeros:, 0,,
) (, ) (, 0) ( 0, ) (, ) f ( ) undefined 4) ( ) 4 lim lim removable discontinuity 6) Neither even nor odd f 5) lim 0 ; lim lim lim 4 jump discontinuity 7) Step : ( ) Step : 4 f undefined 4 0 ( ) - -0-00 - -999-99 -9 f() -0-00 -000 000 00 0 Step : Infinite discontinuity at lim lim 8) 9) < < 0; 0) absolute min of 4 at 0 < <; relative ma of at < < relative min of at ) lim ; lim ) (, ) (, ) (, ) (, ) 4) None 5) (, 6) [ 4, 7) 4 8) rel ma of 7 06 at 45 9) 08 feet per second rel min of 4 at 0 0) ) )
) 4) 5) 5 6) ( ) ( ) 7 g 7) horizontal compression by 4 ; left ; reflection in the -ais 8) 4 9) 8 40) 8 4) 4) 4 4) 7 44) 6 45) ( ) (,, 46) g ( ) 47) even function 48) no 49) 50) [ g]( ) 5) D :, 0 0,, f o [ f o g]( ) ( ) ( ) ( ) D : [, 5 4 5) 5) D : [ 6, ; R : [ 0, 54) 6 55) D : [ 0, ); R : [ 6, 56) passes the HLT 57) currency in U S dollars ( ) currency in Euros 58) 59) D : (, ; R : [ 5, f
CHAPTER REVIEW PROBLEMS (All Sections) Answer the following questions about the power function 5 5, 5, by 5,0,5 Recommended Viewing Window: [ ] [ ] ) State the domain using interval notation ) State the range using interval notation ) State the -intercept 4) State the y-intercept 5) State the end behavior using limit notation 6) Identify the interval(s) over which the function is continuous 7) State the interval(s) over which the function is increasing 8) State the interval(s) over which the function is decreasing Solve each equation Be sure to indicate if any roots are etraneous 9) ( ) 8 0) 5 Solve each inequality Epress the solution set using interval notation ) ( 9 4 ) 5 ) 6 9 < 0 Answer the following questions using the function 5 4 ) Describe the end behavior using limit notation 4) State the total possible number of zeros 5) State the maimum number of turning points 6) Find the zeros by factoring 4 Answer the following questions using the function ( ) ( ) 7) Describe the end behavior using limit notation 8) Determine the real zeros and state the multiplicity of any repeated zeros 9) Sketch the graph completely using long division if ( 9 5) 4 0) Factor 9 4 6 40 ) Divide using synthetic division: ( 4 5 7 ) ( ) is a factor 5 4 ) Find the remainder when 8 7 4 is divided by Is a factor? ) Find the value of k so that the quotient has a remainder of 7 : 7 k 5 4) Use the graphing calculator to solve the following polynomial equation Indicate the multiplicity 4 4 9 0 Window: [ 6, 6, ] by[ 40,40, 0]
Answer the following questions using the function g ( ) 6 9 4 Recommended Window: [ 6, 6, ] by[ 40, 40,0] 5) List the possible rational zeros 6) Determine which, if any, are zeros 7) Write a polynomial function of least degree with integer coefficients in standard form that has zeros at, 5 i, and 5 i Answer the following questions using the function ( ) 4 4 6 Recommended Viewing Window: [ 6, 6, ] by[ 80,60,0] 8) Write ( ) 9) List all the zeros of 0) Write as the product of linear factors f f as the product of linear and irreducible quadratic factors State the domain Determine the equations of any asymptotes and the coordinates of any removable discontinuities (holes) Find the intercepts 5 ) ) 4 Domain Domain VA Hole VA Hole HA OA HA OA y-intercept zero(s) y-intercept zero(s) Solve each equation Be sure to indicate if any of the roots are etraneous ) 0 4) 7 6 8 Enter the function 0 5 into the graphing calculator and answer the following questions Use the standard viewing window, zoom 6 5) Solve: 0 5 0 6) Solve: 0 5 < 0 Solve each polynomial inequality by sketching the graph of the related function 7) 0 5 4 8) 5 4 0 < 0 Solve the rational inequality by using sign analysis 9) 0
CHAPTER ANSWERS ) (, ) [, 0 ) 0 4) 0 5) lim ; lim 6) (, 7) ( 0, 8) (, 0) 9) 6, 0 0) 8, ( is etraneous) ) [ 5, ) [, 5) lim ) 4) 5 5) 4 lim lim 6) 0,,, i 7, i 7 7) 8) lim 9) 0) ( 5)( 7)( )( ) 9) 4,, 5, 5 4 ) 4 5 ) Remainder 0 ; is a factor ) k 8 4) 4, 5, ( M ) p 4 5) 7 7 7, q ±, ±, ±, ± 6, ±, ±, ± 7, ±, ±, ± 6, ± 4 ± 6), 7, 7) 9 9 4 8) ( 4)( )( 6 4) ( M ) 0) ( 4)( ) ( ( 5 )( ( 5 ) 0 ( M); 4 ( M ); ) ) Domain (, ) (, ) (, VA Hole (, ) Domain VA (, ) (, Hole none HA y OA none HA none y OA y-int ( ) 0, 5 zero(s) ( 5, 0) y-int ( 0, 4) zero(s) (, 0) (, 0) ) 5 ± 4), ( is etraneous) 6 5) 5 6) (, 5) ( 5, 7) (, 8] [, 8) (, 5) (, ) 9) [ 5, )
CHAPTER REVIEW PROBLEMS (Sections,,, & 4) b ) Give an eample of a value of b for which ( ) 5 represents eponential growth ) Graph ( ) ( ) 4 f with the graphing calculator Window: 6 6; scl ; 6 y 6; yscl State the domain, range, y-intercept, asymptote, end behavior, and intervals over which the function is increasing or decreasing Dom ain Range y-intercept Asym ptote ) Describe how to transform the graph of g ( ) into the graph of ( ) End Behavior 4) Describe how to transform the graph of ln into the graph of g ) ln( ) ( Increasing Decreasing ( ) g ( ) log is a transformation o log 5) Sketch the graphs of and ( ) f g 6) State the domain of g () 7) State the range of g () 8) State the equation of the asymptote of g () 9) Epand: log 4 7 0) Condense: ln 7 ln a ln b ) Write log 4 6 in eponential form ) Write 5 64 in logarithmic form Evaluate each epression without a calculator ) ln 7 e 4) 6) log 7) 6 Find the domain of each logarithmic equation 9) f ) log( 4 ) log 5 5 5) log log 8 8) ( ) 7ln e ln ( 0) h ( ) ln( 5) Find the inverse of each equation ) ( ) log5 ( ) f ) e e
Solve each eponential equation Round to four decimal places, if necessary ) 5 5 4) 5( ) 0 5) e 4 4 6) e e 0 Solve each logarithmic equation Round to four decimal places, if necessary Check for etraneous roots 7) log ( ) 8) log8 ( ) log8 9 9) ln( ) ln( ) ln( ) 0) ( 4) log ( ) log ) Find the value of log 8 5 using the change of base formula Round to the nearest ten-thousandth ) If $000 is invested at 65%, how much more money would you have after years if the interest was compounded continuously as opposed to monthly? ) George Owens wants to deposit his inheritance into an interest-bearing account that compounds continuously What rate does he need to invest at in order to double his inheritance in 5 years? 4) The number of children, f (), infected by a virus can be modeled by the function 4 9 ln, where is the number of days since the first child was infected After how many days will 40 children be infected? In 000, the world s population was about 6 billion If the world s population continues to grow at a 00t constant rate, the future population P, in billions, can be predicted by P 6e, where t is the time in years since 000 5) Does this model accurately predict our current population? Eplain 6) Some eperts have estimated that the world s food supply can support a population of, at most, 8 billion According to this model, in what year will the world s population reach 8 billion? CHAPTER ANSWERS ) b > 5 ) Reflect over the -ais 4) Reflect over the y-ais compress vertically by epand horizontally by ) Domain, translate one unit right translate one unit down Range y-intercept Asymptote End Behavior Increasing Decreasing ( ) (, 4) ( 0, ) y 4 lim 4 lim never (, 5) 6) (, g() f () 7) (, 8) 9) 4log log ( 7 ) log 0) 7 b ln a ) 4 6
) log 5 64 ) 7 4) 9 4 5) 0 6) 5 7) 8) 9 9) (, 4) 0) (, 5) (, ) ( ) 5 f ) ( ) ln( ) f ) 4) 69 5) 0 99 6) 0 69 7) 8) ± 9 9), 4 ( etraneous ) 0) 4 ) 0 ) $76 ) 77% 4) after about days 5) According to this model, the current world population should be about 86 billion people According to http://wwwworldometersinfo/world-population/ the current world population is 76 billion Therefore, the model overestimates 6) The year 055 Study Tip Studying for 0-50 minutes at a time (with 0 minute breaks in between) is the most effective way to retain information KNOW YOUR TRANSFORMATIONS! Fill in the rules for the transformations of y c translation of c units y f ( c) y y c translation of c units y f ( c) ( ) reflection over the y f ( ) y f y c, c > vertical epansion of f ( c), c > y c, 0 < c < vertical compression of f ( c), 0 < c < translation of c units translation of c units reflection over the y horizontal compression of y horizontal epansion of