Advanced Functions Final Exam Review 2025 PDF

Advanced Functions Page 1 of 6 Final Exam Review Name: 1) Given the following function f ( x)  2( x  1)( x  2)( x  1) 2 , i) Determine the intercep...

Advanced Functions Page 1 of 6 Final Exam Review Name: 1) Given the following function f ( x)  2( x  1)( x  2)( x  1) 2 , i) Determine the intercepts; ii) State the degree of the function; iii) Sketch the graph; iv) Describe the roots of graph. 2) Perform the following divisions and express your answers in the form f ( x)  d ( x) q ( x)  r ( x) a) (3x3  11x 2  7  21x)  (2  3x) b) (6 x5  3x 3  7 x 2  4)  ( x  1) 3) If – 2 is one root of x 2  kx  6  0 , find the other root and the value of k. 4) f ( x)  x3  kx 2  hx  12. When f (x) is divided by x  1 , the remainder is 5. When f (x) is divided by x  3 , the remainder is – 27. Find the values of k and h. 5) Solve x 3  x 2  x  15  0 6) Solve 2 x 3  3x 2  18 x  5  0 7) Solve 8( x  x 2 )  12  ( x 2  x) 2. x  61  8) For the curve y  3   5 ,  62  a) there is a ___________________ asymptote which the equation is ________________. b) there is an _______ - intercept, which is ____________. c) the function is ________________________ (increasing, decreasing, not changing) d) the domain is _________________. e) the range is _________________. 9) Simplify. Express final answer with positive exponents. (3x 3 y 2 ) 4 2 3 2 x y 1 5 Advanced Functions Page 2 of 6 Final Exam Review Name: 10) A culture has 750 bacteria. The number of bacteria doubles every 5.25 hours. How many bacteria (to the nearest hundred) are in the culture after 18.15 hours? 11) Write an exponential function for the following graph: 1 12) In eight days a certain amount of Vanadium-48, V48, decays to of its original amount. What is the half-life 2 of V48? 13) In 1947 an investor bought Van Gogh’s painting Irises for $84 000. In 1987 she sold it for $49 000 000. What annual rate of interest corresponds to an investment of $84 000 which grows to $49 million in 40 years? (1 decimal place) 14) For the curve y  4 log3 ( x  2)  5 a) there is a __________________ asymptote which the equation is _____________. b) there is a ______ - intercept, which is ______________. (2 decimal places) c) the function is ____________________ (increasing, decreasing, not changing). d) the domain is ______________. e) the range is _____________. 15) Write 54  625 in logarithmic form. 2 1 16) Write loga x  loga y  loga z as a single logarithm. 3 4 17) If log 6  m and log 5  n , write log 7.2 as an expression in m and n. Advanced Functions Page 3 of 6 Final Exam Review Name: 18) Use the logarithmic properties to evaluate log 2 32. 19) Use logarithms to solve x. (2 decimal places) a) 7 3x  4 b) 7 3 x  82 x  16  20) If log3 4  1.26 , use logarithms to find an approximation for log3  . Show at least one step. 3 21) Solve 3 log x  log 512  log 8 22) Determine the point of intersection of the curves y  log10 ( x  2) and y  1  log10 ( x  1) 23) Evaluate each of the following by expressing in terms of a related acute angle. 45 7  5  17 a) tan b)  sec c) sin    d) csc 4 6  6  4 5𝜋 24) A wheel has a diameter of 90 cm. If has rotated through an angle of 4 radians. How far will the wheel have travelled? 25) Determine the function for the given cosine graph.   26) Sketch y  2 sin 4 x    8 1   27) For y  cos 2 x    4 , 2  6 Sketch the function for 0    2 and highlight 1 cycle in a proper size and scale. (State the Amplitude, Period, Phase Shift and Vertical translations) 28) When comparing the graphs of y  tan   b and y  cot   b , In what ways they are similar? (In terms of the amplitude, period, …etc) (  3) 29a) Sketch one cycle of y  2 csc 2 4 6 (State the Amplitude, Period, Phase Shift and Vertical translations) b) State the domain and range. Advanced Functions Page 4 of 6 Final Exam Review Name:   8  30) Express and find the exact value of cot   using the correlated acute formula.  3     3  sin x   tan  x   31) Simplify:  2   2  cos(  x)  tan(  x) 13 32) Find the exact value of sin 12 1  33) If sin B  ,  B   , find the exact value of sin 3B 4 2  34) If sec A  2 ,  A   , find the exact value of 2   a) cos A  . b) cos 4 A  4 35) Prove the following identities. 1  sin x  cos x x a) cos x  cos 2 x  cos 3x  cos 2 x(1  2 cos x) b)  cot 1  sin x  cos x 2 36) Simplify and state the restrictions if any. a2  a  2 2a 3a  2   6a 3  a 2  12a  5 3a 2  5a 2a 2  11a  5 x3 1 37) Sketch y  x 2  2x  Simplify the function to determine if there are any holes in the graph.  Determine the domain of the function.  Determine the x and y intercepts.  Determine if there are any vertical or horizontal or oblique asymptotes.  Determine the behaviour of f(x) near the asymptotes by creating a table of values.  Determine cross-overs, if any.  Sketch and label the function with a proper scale. 38) Solve the following rational inequalities. State the answers in interval notations. 2 x 2  5 x  25 0 4  x2 Advanced Functions Page 5 of 6 Final Exam Review Name: 39) At a small clothing company, the estimated average cost function for producing a new line of jeans is x 2  4 x  20 C ( x)  , where x is the number of pairs of jeans produced, in thousands. C(x) is measured in x dollars. a) Calculate the average cost of a pair of jeans at a production level of 3000 pairs. b) Estimate the rate at which the average cost is changing at a production level of 3000 pairs of jeans. 1 40) If f ( x)  x 2  81 , g ( x)  x 4 , h( x)  , k ( x)  2  3x. 2x a) ( f  g )( x) b) Domain of (h  k )( x) c) ( g  f )( x) d) (h  h)( x)   5  e) g  h   f) Symmetry of f 1  f ( x) g) k 1 (k ( g ( x))) h) k ( x)  h( x)   2  41) If f ( x)  4 x  3 and h( x)  4 x 2  21 , find a function g such that f  g  h. 42) If f ( x)  x 2  2 x  3 , where x  A  {x | 5  x  2} , and g ( x)  6 x  18 , where x  B  {x | 3  x  9} , g ( x) find the functions and its domain of. f ( x) ============================================================================ Additional Exam reviews: Unit reviews at the end of each units in the text book Chapter 1: P. 74 #1 – 18 Chapter 2: P. 140 #1 – 14, 17, 18 Chapter 3: P. 192 #1 – 11, 12ab, 13, 15, 16 Chapter 4: P. 244 #1 – 18, 20, 22, 23 (plus review the trig identities worksheet #1-36) Chapter 5: P. 300 #1 – 13, 14abc Chapter 6: P. 356 #1 – 9, 11 – 18 Chapter 7: P. 408 #1 – 14, 16 – 17 Chapter 8: P. 472 #1, 2, 4 – 13 Advanced Functions Page 6 of 6 Final Exam Review Name: Reference Answers: 1) i) x-int: – 1, – 2,1 y-int: – 4 ii) 4th iii) see graph below iv) 2 diff & 2 equal real roots 2a) f ( x)  (3x  2)( x 2  3x  5)  3 b) f ( x)  ( x  1)(6 x 4  6 x3  3x 2  4 x  4)  8 3) k = –1, x = – 2, 3 4) k = – 3, h = – 5 5) x = 3 6) x = 2.5 or  2  3 7) x = – 1, – 2, 2, 3  8 x 58 / 5 8a) horizontal, y = – 5 b) y, – 2 c) decreasing d) x  R e) y > – 5 9) 81y 7 10) Approx 8200 11) y  4(2 x )  2 12) 16 days 13) Approx 17.3% 14a) vertical, x = 2 b) x, 2.25 c) increasing d) x > 2 e) y  R 15) log5 625  4 x2 / 3 16) log a 17) 2m – n 18) 10 19a) 2.29 b) 0 y1 / 4 z 20) 1.52 21) 4 22) (4, log10 2 ) 2 1 225𝜋 23a) 1 b) c) d) 2 24) cm 25) y  2 cos 2( x  0.5)  3 3 2 4 26) see graph below 27) see graph below 28) same midline, period, zeros of one are the asymptotes of the other. Shape is similar, reflected and translated. 1 29a) see graph below b) D: x  3  3k , k  I R: y  6 or y  2 30) 3  3 1 11  1 3 1 31) csc 2 x 32) 33) 34a) b) 2 2 16 2 2 2 a 2  3a  14 1 5 36) , a  5,1, ,0, 2(2a  1)(a  5) 2 3 37) VA 0 & -2 , OA y = x – 2 x-int = 1 y-int: none , cross over x = ¼ , see graph below 17 38) (,5], (2,2), [2.5, ) 39a) b) $  1.2222 /pair 3 2 1 2  3x 40a) x 8  81 b) x  c) x 4  162 x 2  6561 d) x e) f) odd g) x 4 h) 3 625 2x 9 6 41) g ( x)  x 2  42) , x  1,3  x  2 2 x 1 #26 #29 #1iii #37 #27