Final Eam Review Sheet Algebra for Calculus Fall 007 Find the distance between each pair of points A) (,7) and (,) B), and, 5 5 Find the midpoint of the segment with endpoints (,) and (,) Find each of the following: f () f ( ) f (0) For how many values of does f ( ) Approimate these values of Find an equation of the circle having diameter with endpoints (, 5) and (7,) center (, 5) and passing through (,7) Find the center and radius of the circle + y + 0y + 0 5 Determine whether each of the following defines y as a function of Identify domain and range {(,7),(, 7),(7, ),(0,),(, )} e) y y y y A graph of a function, f, is shown (-,) - - - 7 Given that f ( ) f (0) f ( ) f ( ) e) f ( + h) f) f ( + h) h +, find: 8 Find the domain of each of the following: + g( ) + 5 h( ) + j( ) + 5 e) k( ) ln( + ) 9 Find the domain and range of each of the functions represented by the following graphs 0 According to the US Bureau of the Census, workers compensation payments in Florida rose from $ million in 980 to $97 billion in 990 Find the average rate of change in payments A KSU football player ran for 05 yards in 000 and for 978 yards in 00 Find the average rate of change in his performance # 7: Write an equation of the line: through (, ) and (, ) - - (,-) through (0,) and (, )
Final Eam Review Sheet Algebra for Calculus Fall 007 parallel to + y and passing through the point (,) 5 perpendicular to + y and through (, ) with undefined slope that passes through (-5,0) 7 with zero slope that passes through (, ) 8 If a town starts with an initial population of 00000 and 50 people leave each year, what is the linearequation that relates the population (P) to the number of years since the town began (n)? 9 The table below shows the average price, in dollars, of a personal computer at Moratech Inc in several recent years Year, Average Price P 990, 0 50 99, 50 99, 50 999, 9 05 00, 800 Model the data with a linear function using years and 9 and predict the average price in 005 Sketch the graph of f as defined in # 0 above Using the graph below, determine the intervals on which the function is ( increasing, ( decreasing, and ( constant A farmer has 0 yards of fencing with which to enclose two adjacent rectangular plots, one for corn and the other for soybeans Suppose the width of each plot is Epress the total area of the two plots as a function of Find the domain of the function Consider the functions F and G as shown in the following graph Find the domain of F, the domain of G, and the domain of F + G Make a sketch of the graph of F + G Make a sketch of the graph of F G 5 Given the functions f and g defined by the following: and g( ) Find the domain of f + g Find f + g Given the functions f and g defined by the following: + and g( ) Find the domain of f + g Find f + g 7 For + and g( ) Find ( f g)( ) and ( g f )( ) Find the domain of ( f g)( ) and ( g f )( ) 8 Given that + and g( ) + 5, find ( f g)( ) and ( g f )( ) 0 Given the function f defined piecewise as, for, +, for < <, follows:, for Find F G 9 Determine whether the graph of y is symmetric with respect to the -ais, the y-ais, or origin f (5) f ( ) f (5)
Final Eam Review Sheet Algebra for Calculus Fall 007 0 Test algebraically whether the following functions are even, odd, or neither f ( ) + g( ) h ( ) + Write an equation for a function that has the shape of y but shifted right units and down units y reflected through the - ais and shifted up 5 but stretched vertically by a factor of, y but shrunk horizontally by a factor of and shifted up units Given the graph below of a function, f -0-5 5 0 Sketch the graph of f ( ) + f ( + ) ( ) f f ( ) - - - Epress in terms of i : 9 Simplify (5 + i)( + i) ( i) (8 + i) i + i 5 Find the zeros of f ( ) + 7 g ( ) + 5 + 8 0 8-0 -5 5 0 - i 8 Complete the square on the function f given by + Name the verte of f Name the ais of symmetry Determine if there is a maimum or minimum function value and find that value 7 A ball is thrown directly upward from a height of 0 feet with an initial velocity of 0 ft/sec If the height of the stone t seconds after it has been thrown is given by the function s( t) t + 0t + 0, determine the time at which the ball reaches its maimum height and find the maimum height 8 The graph below represents a polynomial function Is the degree of the leading term even or odd? 9 Name the zeros and their multiplicities of g( ) ( + ) ( )( + ) Then indicate whether the graph would cross or be tangent to the - ais at each zero Also indicate the end behavior and eplain your reasoning 0 Sketch a graph by hand of h( ) ( )( + ) ( ) Be sure to show all steps, including naming the zeros, multiplicities, end behavior, and showing the sign chart Use synthetic division to divide P( ) + 0 by D( ) and write the quotient and remainder Use the Factor Theorem to determine if + is a factor of P( ) + 8 + 7 List all the possible rational zeros of each of the following: f ( ) 0 + h ( ) + 7 + Find all the zeros of these: 8 + 8 g ( ) + h ( ) + 9 0 m ( ) 7 + 8 9 + 7 5 Factor each of the following into linear factors: p( ) 0 7 7 + q( ) 5 8 + 5 r( ) + + - -
Final Eam Review Sheet Algebra for Calculus Fall 007 Find a polynomial function of degree with the following zeros:, 0, 8, i, i 5,, + 7 Suppose that a polynomial function of degree 5 with rational coefficients has the following numbers as zeros Find the other zeros, + 5, 5 i 0, 5,, 0i 8 Name the vertical and horizontal asymptotes of the following rational functions h( ) 8 m( ) + 5 + n( ) + 5 + p( ) + 5 9 Find the zeros of each of the functions in # 8 < 0 5 + + e) 5 Which of the following functions are one-to-one? + 7 +, ( ) 5 Find f for each of these 5 + + + 8 ( + ) 5 Verify that and g( ) + are inverses of each other by using composition of functions 5 Sketch the inverse of the function represented by each graph below 5 Without using your graphing calculator, sketch a graph of each of these f ( ) + e 55 How much money will be in an account in 0 years if it compounds 75% interest quarterly and you deposit $5,000 now? 5 Evaluate each of these log log log 8 log8 e) log7 9 50 Solve each of the following inequalities: -0-5 5 0 57 Write in logarithmic form: r e s 8 + 0 + + 8 5 - - - -8 58 Write in eponential form: logc d ln p q 59 Find the domain of ln( 5)
Final Eam Review Sheet Algebra for Calculus Fall 007 5 0 Epress in terms of sums and differences of logarithms ln y z ln e Write as a single logarithm: ln ( 9 ) ln( + ) + ln( + y ) Simplify log 50 log p p Solve each of these log log ( ) + log ( + ) 5 + e 0 e) ln ln( ) ln log a ln 8 a b 5 m n e The population of a country doubled in 8 years What was the eponential growth rate? 5 Suppose $8,000 is invested at rate k, compounded continuously, and grows to $, in years Find log 5 using the change of base formula 7 Average cell phone prices have fallen sharply since their introduction to the market in 98 In 98, the average price was $95, and in 00, it was only $5 Assuming the average price of a cell phone decreased according to the eponential model: Find the value of k and write an eponential function that describes the average price of a cell phone after time t, in years, where t is the number of years since 98 Estimate the price of a cell phone in 00 At what time t was the price half the original price? 8 Evaluate P 8 C 9 Epand ( ) ( ) 5 n + ( + y) 5 70 Find the fourth term of the binomial epansion + y of ( ) Find the interest rate Find the eponential growth function Find the balance after 0 years Find the doubling time
Final Eam Review Sheet Algebra for Calculus Fall 007 Find the distance between each pair of points A) (,7) and (,) B), and, 5 5 Find the midpoint of the segment with endpoints (,) and (,) Find each of the following: f () f ( ) f (0) For how many values of does f ( ) Approimate these values of Find an equation of the circle having diameter with endpoints (, 5) and (7,) center (, 5) and passing through (,7) Find the center and radius of the circle + y + 0y + 0 5 Determine whether each of the following defines y as a function of Identify domain and range {(,7),(, 7),(7, ),(0,),(, )} e) y y y y A graph of a function, f, is shown (-,) - - - 7 Given that f ( ) f (0) f ( ) f ( ) e) f ( + h) f) f ( + h) h +, find: 8 Find the domain of each of the following: + g( ) + 5 h( ) + j( ) + 5 e) k( ) ln( + ) 9 Find the domain and range of each of the functions represented by the following graphs 0 According to the US Bureau of the Census, workers compensation payments in Florida rose from $ million in 980 to $97 billion in 990 Find the average rate of change in payments A KSU football player ran for 05 yards in 000 and for 978 yards in 00 Find the average rate of change in his performance # 7: Write an equation of the line: through (, ) and (, ) - - (,-) through (0,) and (, )
Final Eam Review Sheet Algebra for Calculus Fall 007 parallel to + y and passing through the point (,) 5 perpendicular to + y and through (, ) with undefined slope that passes through (-5,0) 7 with zero slope that passes through (, ) 8 If a town starts with an initial population of 00000 and 50 people leave each year, what is the linearequation that relates the population (P) to the number of years since the town began (n)? 9 The table below shows the average price, in dollars, of a personal computer at Moratech Inc in several recent years Year, Average Price P 990, 0 50 99, 50 99, 50 999, 9 05 00, 800 Model the data with a linear function using years and 9 and predict the average price in 005 Sketch the graph of f as defined in # 0 above Using the graph below, determine the intervals on which the function is ( increasing, ( decreasing, and ( constant A farmer has 0 yards of fencing with which to enclose two adjacent rectangular plots, one for corn and the other for soybeans Suppose the width of each plot is Epress the total area of the two plots as a function of Find the domain of the function Consider the functions F and G as shown in the following graph Find the domain of F, the domain of G, and the domain of F + G Make a sketch of the graph of F + G Make a sketch of the graph of F G 5 Given the functions f and g defined by the following: and g( ) Find the domain of f + g Find f + g Given the functions f and g defined by the following: + and g( ) Find the domain of f + g Find f + g 7 For + and g( ) Find ( f g)( ) and ( g f )( ) Find the domain of ( f g)( ) and ( g f )( ) 8 Given that + and g( ) + 5, find ( f g)( ) and ( g f )( ) 0 Given the function f defined piecewise as, for, +, for < <, follows:, for Find F G 9 Determine whether the graph of y is symmetric with respect to the -ais, the y-ais, or origin f (5) f ( ) f (5)
Final Eam Review Sheet Algebra for Calculus Fall 007 0 Test algebraically whether the following functions are even, odd, or neither f ( ) + g( ) h ( ) + Write an equation for a function that has the shape of y but shifted right units and down units y reflected through the - ais and shifted up 5 but stretched vertically by a factor of, y but shrunk horizontally by a factor of and shifted up units Given the graph below of a function, f -0-5 5 0 Sketch the graph of f ( ) + f ( + ) ( ) f f ( ) - - - Epress in terms of i : 9 Simplify (5 + i)( + i) ( i) (8 + i) i + i 5 Find the zeros of f ( ) + 7 g ( ) + 5 + 8 0 8-0 -5 5 0 - i 8 Complete the square on the function f given by + Name the verte of f Name the ais of symmetry Determine if there is a maimum or minimum function value and find that value 7 A ball is thrown directly upward from a height of 0 feet with an initial velocity of 0 ft/sec If the height of the stone t seconds after it has been thrown is given by the function s( t) t + 0t + 0, determine the time at which the ball reaches its maimum height and find the maimum height 8 The graph below represents a polynomial function Is the degree of the leading term even or odd? 9 Name the zeros and their multiplicities of g( ) ( + ) ( )( + ) Then indicate whether the graph would cross or be tangent to the - ais at each zero Also indicate the end behavior and eplain your reasoning 0 Sketch a graph by hand of h( ) ( )( + ) ( ) Be sure to show all steps, including naming the zeros, multiplicities, end behavior, and showing the sign chart Use synthetic division to divide P( ) + 0 by D( ) and write the quotient and remainder Use the Factor Theorem to determine if + is a factor of P( ) + 8 + 7 List all the possible rational zeros of each of the following: f ( ) 0 + h ( ) + 7 + Find all the zeros of these: 8 + 8 g ( ) + h ( ) + 9 0 m ( ) 7 + 8 9 + 7 5 Factor each of the following into linear factors: p( ) 0 7 7 + q( ) 5 8 + 5 r( ) + + - -
Final Eam Review Sheet Algebra for Calculus Fall 007 Find a polynomial function of degree with the following zeros:, 0, 8, i, i 5,, + 7 Suppose that a polynomial function of degree 5 with rational coefficients has the following numbers as zeros Find the other zeros, + 5, 5 i 0, 5,, 0i 8 Name the vertical and horizontal asymptotes of the following rational functions h( ) 8 m( ) + 5 + n( ) + 5 + p( ) + 5 9 Find the zeros of each of the functions in # 8 < 0 5 + + e) 5 Which of the following functions are one-to-one? + 7 +, ( ) 5 Find f for each of these 5 + + + 8 ( + ) 5 Verify that and g( ) + are inverses of each other by using composition of functions 5 Sketch the inverse of the function represented by each graph below 5 Without using your graphing calculator, sketch a graph of each of these f ( ) + e 55 How much money will be in an account in 0 years if it compounds 75% interest quarterly and you deposit $5,000 now? 5 Evaluate each of these log log log 8 log8 e) log7 9 50 Solve each of the following inequalities: -0-5 5 0 57 Write in logarithmic form: r e s 8 + 0 + + 8 5 - - - -8 58 Write in eponential form: logc d ln p q 59 Find the domain of ln( 5)
Final Eam Review Sheet Algebra for Calculus Fall 007 5 0 Epress in terms of sums and differences of logarithms ln y z ln e Write as a single logarithm: ln ( 9 ) ln( + ) + ln( + y ) Simplify log 50 log p p Solve each of these log log ( ) + log ( + ) 5 + e 0 e) ln ln( ) ln log a ln 8 a b 5 m n e The population of a country doubled in 8 years What was the eponential growth rate? 5 Suppose $8,000 is invested at rate k, compounded continuously, and grows to $, in years Find log 5 using the change of base formula 7 Average cell phone prices have fallen sharply since their introduction to the market in 98 In 98, the average price was $95, and in 00, it was only $5 Assuming the average price of a cell phone decreased according to the eponential model: Find the value of k and write an eponential function that describes the average price of a cell phone after time t, in years, where t is the number of years since 98 Estimate the price of a cell phone in 00 At what time t was the price half the original price? 8 Evaluate P 8 C 9 Epand ( ) 5 ( ) n + ( + y) 5 70 Find the fourth term of the binomial epansion + y of ( ) Find the interest rate Find the eponential growth function Find the balance after 0 years Find the doubling time