MHF 4U Exam Review - 1

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1 MHF U Eam Review MHF U Eam Review - Part I Polynomial Functions:. Determine the zeroes of the function f ( ) ( 5)( + + ).. Determine the equation of: a. The cubic function with zeroes, -, -5 and y-intercept 0. b. The quartic function with roots at & -, and a double root at, and through (,-0). By calculating the Finite Differences, determine a. Which type of polynomial function best models this data. b. Determine the minimum and maimum number of zeroes and turning points for this type of function f ( ) a) Find the remainder when + 7 is divided by b) Find k if ( 5) is a factor of 5 + k. c) Divide ( y y + y 9) ( y + ). State answer using a correct division statement. 5. Factor fully: a) b) + 6. Solve for, C : a) b) c) 9z z 0 d) 7 6 e) ( ) 7. Solve the following polynomial inequalities. a) t t t > 0 b) t + t t < 0 c) ( )( + )( + 5) 0 8. For the function g( ) ( ) a. State the degree of the function and comment on its end behaviour. b. State the zeroes and 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. Part II Rational Functions +. Find the asymptotes of f ( ) + +. Sketch the graph of f ( ).. I) Given: g( ) ii) Given: h() a. Determine and y intercepts. a. Determine and y intercepts. b. State the domain. b. State the domain. c. State any asymptotes. c. Find all asymptotes. d. Are there any holes? D. Are there any holes?

2 MHF U Eam Review -. 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, and a hole at c. Vertical Asymptote at, horizontal asymptote at y 0, no -intercept and a y intercept at 5. Determine if the following functions have crossover points: a) f( ) ( )( + ) b) f ( ) Solve and state any restrictions. + a. b c The estimate revenue and cost functions for the manufacture of a new product are R()- +5 and C()5+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 8. The value of a car, t years after it is bought, is modeled by V ( t) t a) What will the car be worth in the long run? b) Find the average rate of change in the value of the car between and 5 years and the instantaneous rate of change at years. 9. Sketch f ( ) + 9 and its reciprocal on the same grid. 0. Solve a) ( ) > 0 + b) > 6 Part III Trigonometric Equations and Periodic Phenomena:. Sketch θ of: c) y sin ( θ )- b) y - cos θ + c) y csc θ +. Given f() sin: a) Find the average rate of change (to 5 decimals) of the function from to. What does this tell us about the graph? b) Find the instantaneous rate of change (to 5 decimals) of the function at. What does this tell us 6 about the graph?. State the RAA, principle angle and two co-terminal angles for each of the following. a) b)

3 . Find each value. a) cscθ if sinθ 5 b) cot θ if cscθ 7 MHF U Eam Review - 5. Find each of the following. Use eact values only. 7 a) sin b) sec c) cot 6 d) cos 6. Find all values for 0 θ. a) cos θ b) secθ c) cot θ d) tanθ 7. Solve: θ a) sin θ b) cos c) cos θ + 8. Solve: a) cos b) sin + sin 0 c) 0cos () + 7cos() 6 d) cos () 0 e) tan + tan 0 f) 6cos sin 0 g) cos sin 9. Evaluate the following: 5 a) cos 6 b) sec c) sin d) tan 0. Write as a single trig function. a) sin Acos B cos Asin B b) θ d) sin e) tan tan sin cos f) c) 0 sin cos tan tan tan tan. Develop a formula for sin θ in terms of sinθ.. If sin and cos y, where 0 and y, find the following. 5 a) sin( y) b) tan( + y). If cot for < <, find the following. a) tan b) csc. Prove the following identities. + tan a) csc tan b) cos sin tan cos sin cos c) + sin cos cos sin

4 cos( y) d) + tan tan y cos cos y g) cot θ + tanθ csc θ cos e) sin tan f) cos cos sin sin MHF U Eam Review - 5. a) State the equations of the vertical asymptotes for i) y sec for εr ii) y tan for εr b) State the period of y tan. 6. A ferris wheel, with a diameter of 0m, makes a full rotation in minutes. Passengers board at the bottom of the ferris wheel, which is m above ground. a) Find the equation that best models the height of a passenger. b) Find the height of a passenger who has been riding the ferris wheel for 50 seconds. 7. Find the equations of the following graphs in terms of both sine and cosine: a) b) 5/ / / / / / / 5/ / / / / Part IV - Eponential and Logarithmic Functions:. Sketch the following functions. (a) y ( ) (b) y ( ) (c) y +. Sketch the following functions. (a) y log + (b) y log ( + ) (c) y log( ) +. Epress in logarithmic form (a) 5 5 (b) 6 0 (c) 9 7 (d) 5 5. Epress in eponential form (a) log 0 (b) log 6 ( ) 5. Evaluate the following 6 (c) log 9 7 (a) log 7 9 (b) log 6 (c) log (d) log (f) log (6 6) (e) log 576 log (f) log6 8 + log6 6. Solve (a) log log 5 + log (b) log log log (c) log ( + ) + log ( ) (d) log ( 7 + ) log ( )

5 7. Epress as a single log: (a) log5 + log5 y log5 z (b) [log + log y] [log a + log b] MHF U Eam Review Solve + (a) 5 (b) + ( ) 0 0 (c) 8 6 d) 6( 0 ) A sample of 500 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. 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 depreciates in value by.5% every year. When will the car be worth half its original value?. 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 B.C. with a magnitude of.6 on the Richter Scale. How many times as intense as the 997 earthquake was the 99 earthquake? Answer correct to decimal places.. The loudness level of a heavy snore is 69 db. How many times is this more intense than conversational speech at 60 db? Answer correct to decimal places.. A liquid has a ph of.5. What is the hydrogen ion concentration (mol/l) in the liquid? 5. Given the function y, determine: a. The average rate of change from t5 seconds to t6 seconds. b. The instantaneous rate of change at t5 seconds. Part V Combination of Functions:. Given f() and g() log 5, find the domain of: a) (f + g)() b) (f g)() c) (f/g)(). Given f() + and g(), sketch the graphs of: a) (f - g)() b) (f g)(). Given f() and g(), find: a) f(g()) b) g(f()) c) f(g()) d) g(f()). A spherical hailstone grows in a cloud. The hailstone maintains a spherical shape while its radius increases at a rate of 0. mm/min a) Epress the radius, r, in millimetres, of the hailstone, as a function of the time, t, in minutes. b) Epress the volume, V, in cubic millimetres, of the hailstone, in terms of r. c) Determine (V r)(t) and eplain what it means. d) What is the volume of the hailstone h after it begins to form?

6 Answers: MHF U Eam Review - 6 Part I Polynomial Functions:. 5, -5,. a) f() b) f() a) The th differences are constant (value is ). Therefore it is quartic. b) Minimum number of zeroes is 0, Maimum number of zeroes is, minimum number of turning points is, maimum number of turning points is. a) 5.5 b) 9.6 c) y y + y 9 (y + )(y ) + (6y 5) 5. a) ( + )( )( + ) b) ( ) ( + ) 6. a),, b) ±, c) z ± i 0, d), ± e) ±, 7. a) < t < 0, t > b) 0 < t <, t < c) 5, 0, 8. a) Degree is 5, For end behaviours, as +, y and as, y + b) Zeroes are 0 and c) The function is decreasing when <. (approimately from graph) and when > d) g( ) < 0 when 0 < < and > e) Local minimum at approimately (., -.) and local maimum at (,0) Part II Rational Functions: y 8.0 y+. VA: HA: y I) a) -intercept is and y-intercept is R,, c) VA:, HA: y 0 d) There are no holes. b) Domain D { } II) a) There is no -intercept and y-intercept is b) Domain D { R, ± } c) VA: HA: y 0 d) There is a hole at.. a) y b) y + ( )( )( + ) 5. a) (-, -) b) (0, -) c) y 6 ( + )

7 6. a) 5,, b) or, ± c) ± 5, 0 MHF U Eam Review - 7 ( )( ) 7. a) AP( ) 8 or AP( ) + 0 b) From the factored form, you can find zeros and from the form g( ) + you can find oblique h( ) > 0, R d), asymptotes c) The Domain is: D { } 8. a) Worth $66 in long run 9. b) Average Rate of Change between t and t5 is: -$7.86. Instantaneous Rate of Change at t years is -$60.50 Part III Trigonometric Equations and Periodic Phenomena. a) ARC 0. b)ir C a). a) 8,,, b) 5 0 b) 5 5,,, 5. a) b) - c) d) 6. a), b) 5, c) 7, d), a) + n, + n b) + n c) + n,n a) + n, + n, + n, + n b),, c) + n, + n d) + n, + n, + n, + n e) + n,. 6 + n f),,.87, a) cos b) csc c) d) a) sin (A B) b) tan() c) 5 sin() d) cos θ e) -sin f) -tan g) + n. sinθ sin θ. (a) 6 (b) (a) 5 (b) 5. (a) (i) (n+),n I (ii) (n+),n I (b) Period is 6. h(t) -0 cos t 7. 7 a) y sin + or y cos( ) + b) y or y sin cos

8 Part IV Eponential and Logarithmic Functions:. (a) (b) (c) MHF U Eam Review - 8. (a) (b) (c). (a) log 5 5 (b) log 6 0 (c) log 9 7. (a) 0 (b) 6 6 (c)9 7 (d) log (a) (b) (c) (d) 5 (e) 0 (f) (g) 6 (h) 6 (i) 7 (j) (l) (m) (k) 0 6. (a) 5 (b) 6 (c) 8 (d) (e) 6 (f) 9 7. (a) log y y (b) log z a b t 8. (a). (b) (c) (d) (a) N( t) 500( ) 0 (b) 0.0 min 0. (a) N( t) (b) 9.79 days years (a) mol/l 5. (a) 07 (b) 9.57 Part V Combination of Functions:. a) { > 0, R} b) { > 0, R} c) { > 0,, R} y y 0 0. a) b) t a) 5 b)dne c) f o g 6 d) go f. a) r 0.t b) V /r c) V 0.0/Bt d) 78 mm