a. The cubic function with zeros 1, -2, -5 and y-intercept 10. b. The quartic function with zeros 2, -1, 3 (order 2) and passing through (4,-10)

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1 MHFU Final Eam Review Polynomial Functions Determine the zeros of the function f ( ) ( )( + + ) The remainder when + C is divided by + is 8 Determine the equation of: a The cubic function with zeros, -, - and y-intercept 0 b The quartic function with zeros, -, (order ) and passing through (,-0) By calculating the Finite Differences, determine a Which type of polynomial function best models this relationship b Determine the minimum and maimum number of zeros and turning points for this type of function f ( ) 0 8 a) Without using division (long or synthetic), find the remainder when + 7 is divided by b) is a factor of + k Find k c) Divide using long division ( y y + y 9) ( y + ) Factor fully a) + y b) c) + d) 8 + e) Solve for, C a) b) c) 9z z 0 e) f) 7 g) ( ) 8 Solve for t or a) t t t > 0 b) t + t t < 0 c) ( )( + )( + ) 0 9 Find the equation of a polynomial function in standard form, with the given the roots: a), -, b),, c), ± i d) 0, ±, 0 For the function g( ) ( ) a State the degree of the function and comment on its end behaviour b State the zeros and with the aid of a table of values, sketch the function c State from the graph, the interval(s) where g( ) is decreasing d Find, algebraically where g( ) < 0 Illustrate, in colour, on the graph e State the coordinates of any local/absolute ma/min points

2 MHFU Final Eam Review Rational Functions + Find the vertical and horizontal asymptotes for f ( ) Given: f ( ) Find the domain, intercepts and vertical and + horizontal asymptotes Sketch the graph Given: g( ) + + a Determine and y intercepts b State the domain c State any asymptotes d Are there any holes? Given: 9 a Find the and y intercepts b Find the domain c Find any asymptotes d Are there any holes? Create a function that has a graph with the given features a Vertical Asymptote at and horizontal asymptote at y b Two vertical asymptotes at and, with a hole at c Vertical Asymptote at, horizontal asymptote at y 0, no -intercept and a y intercept at d No vertical asymptote and -intercepts at and Graph f ( ) Find the domain, the intercepts, the location of any asymptotes and describe the function s behaviour near these asymptotes 7 Simplify and state restrictions a + + b 8 Solve and state any restrictions a b + c 0 8r + + r d 0 + e 0 ( + ) f 0 0

3 MHFU Final Eam Review 9 The estimate revenue and cost functions for the manufacture of a new product are R()- + and C()+8 P( ) a) Epress the average profit function AP( ), in two different forms b) Eplain what can be determined from each form c) What is the domain of the function in this contet? d) What are the break-even quantities? t 0 The value of a car, t years after it is bought, is modeled by V ( t) t a) State the domain of V(t), determine any intercepts as asymptotes, and sketch the function b) What will the car be worth in the long run? c) After how many years will the car be worth $00? d) Find the average rate of change in the value of the car between and years and the instantaneous rate of change at years Sketch f ( ) + 9 and its reciprocal on the same grid Trigonometry Review Sketch θ of: a) y sin θ - b) y - cos θ + c) y sin θ + d) y - cos ( θ + ) - Sketch f(θ ) sin θ + on the interval, a) Find the average rate of change (to decimals) of the function from to What does this tell us about the graph? b) Find the instantaneous rate of change (to decimals) of the function at What does this tell us about the graph? State two co-terminal angles for each of the following Your answers must include a) b) c) d) Find each function value a) cscθ if sinθ b) cot θ if cscθ 7 Find each of the following Use eact values only

4 MHFU Final Eam Review 7 a) sin b) sec c) cot d) cos Find all values for 0 θ a) cos θ b) secθ c) cot θ d) tanθ 7 Solve for 0 θ θ a) cos θ b) sin θ c) tan θ d) cos e) sin θ f) cos θ + 8 Solve for 0 a) cos b) sin + sin 0 c) 0cos () + 7cos() d) cos () 0 e) tan f) tan + tan 0 g) cos sin 0 h) cos sin i) sin -cos + 9 Write as a trig function of the related angle and evaluate: a) cos b) tan 0 Write as a trig function of the co-related (co-function) angle and evaluate a) sec b) sin Write as a single trig function a) sin Acos B cos Asin B b) cos Acos A sin Asin A c) tan tan d) cos M sin M e) 0 sin cos f) sin θ

5 MHFU Final Eam Review g) sin cos h) cos + θ i) tan a cos a j) tan tan tan tan Evaluate the following a) sin cos - cos sin c) tan + tan tan tan e) cos - sin g) sin i) tan b) cos cos sin sin d) cos + sin f) (cos + sin ) h) cos 8 j) sin Simplify a) (sin + cos )(sin cos ) + sin b) tan sin sec c) sin( 0 ) + cos(0 ) sin d) sin a cos a e) sin a cos a + f) ( sin + cos ) sin g) sin cos sin cos Develop a formula for sin θ in terms of sinθ If sin and cos y, where 0 and y, find the following a) sin( y) b) tan( + y) c) sec( y) If cot for < <, find the following a) tan b) cos c) csc d) sin e) tan

6 MHFU Final Eam Review 7 Prove the following identities + tan a) csc tan b) cos sin tan cos cos sin c) + sin cos cos sin d) cos( + y)cos( y) cos sin y cos( y) e) + tan tan y cos cos y f) csc + sin( ) + csc( ) sin cot + cos g) sin tan i) sin + + sin cos h) cos cos sin sin j) cot θ + tanθ csc θ k) sin t sect sin t cos t cost 8 Sketch y cot for 9 a) State the equations of the vertical asymptotes for i) y sec for εr ii) y tan for εr b) State the local minimums for y csc for εr c) State the period of y tan Eponential and Logarithmic Functions Sketch the following (a) y ( ) (b) y ( ) (c) y ( ) (d) y (e) y + (f) y (g) y + Sketch the following (a) log (b) y log + (c) y log( ) (d) y log ( + ) (e) y log( ) (f) y log + g) y log( ) +

7 MHFU Final Eam Review Epress in logarithmic form (a) (b) 0 (c) 9 7 (d) Epress in eponential form (a) log 0 (b) log ( ) (c) log 9 7 Evaluate the following (a) log 7 9 (b) log (c) log 7 (d) log (e) log 7 (f) log 8 (g) log 8 (h) log7 7 (i) log 9 7 (j) log ( ) (k) log (l) log 7 log (m) log 8 + log Solve (a) log log + log (b) log log log (c) log log log (d) log ( + ) + log ( ) (e) log log (f) log ( 7 + ) log ( ) Epress as a single log: (a) log + log y log z (b) [log + log y] [log a + log b] 8 Solve (a) (b) + ( ) 0 0 (c) ( ) 0 + (d) 8 (e) ( 0 ) A sample of 00 cells in a medical research lab doubles every 0 min a) Determine a formula for the number of cells at time t, where t is measured in minutes b) How long will it take for the population to reach 8 000? Answer correct to decimal places 0 In 987 there were about cell phone users in Canada In 999, there were about 0 million cell phone users What is the percent increase per year? Answer correct to decimal places A sample of 700 cells in a medical research lab triples every 0 min a) Determine a formula for the number of cells at time t b) How long will it take for the population to reach 8 000? Answer correct to decimal places 7

8 MHFU Final Eam Review A sample of radioactive iodine- atoms has a half-life of about 8 days Suppose that iodine- atoms are initially present a) Determine a formula for the number of atoms at time t, where t represents number of days b) How long will it take for the sample to reach atoms? Answer correct to decimal places A new car costs $ 000 In years it will be worth $900 What is the rate of depreciation per year? Answer in percent, correct to decimal places Most of Canada s earthquakes occur along the west coast In 99, there was an earthquake in the Queen Charlotte Islands that had a magnitude of 8 on the Richter Scale In 997 there was an earthquake in south-western BC with a magnitude of on the Richter Scale How many times as intense as the 997 earthquake was the 99 earthquake? Answer correct to decimal places The loudness L in decibels ( 0 of a bel) of a sound of intensity I is defined to be L 0log I I 0 where I 0 is the minimum intensity audible to the human ear a) The loudness level of a heavy snore is 9 db How many times is this more intense than conversational speech at 0 db? Answer correct to decimal places b) Sound is times less intense if earplugs are worn What would the decibel level of snoring be if earplugs were worn? Answer correct to the nearest db Given the function y, determine: a the average rate of change from t seconds to t seconds b the instantaneous rate of change at t seconds 8