Math 30-1 Review Package
January 11, 2017 | Author: Marc Lambert | Category: N/A
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Math 30 – 1 – Final Exam Review Booklet
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Chapter 1 (Transformtions) Review Multiple Choice For #1 to #6, choose the best answer. 1. The graph y f (x) contains the point (3, 4). After a transformation, the point (3, 4) is transformed to (5, 5). Which of the following is a possible equation of the transformed function? A B C D
y 1 f (x 2) y 1 f (x 2) y 1 f (x 2) y 1 f (x 2)
2. The graph of y x is transformed by a vertical stretch by a factor of 3 about the x-axis, and then a horizontal translation of 3 units left and a vertical translation up 1 unit. Which of the following points is on the transformed function? A B C D
5. Given the graph of y f (x), what is the invariant point under the transformation y f (2x)?
(0, 0) (1, 3) (3, 1) (3, 1)
3. The graph of y x is vertically stretched by a factor of 2 about the x-axis, then reflected about the y-axis, and then horizontally translated left 3. What is the equation of the transformed function? A y 2 x 3
1 ) 2
A (1, 0)
B (0,
C (1, 1)
D (3, 1)
6. What will the transformation of the graph of y f (x) be if y is replaced with y in the equation y f (x)? A B C D
It will be reflected in the x-axis. It will be reflected in the y-axis. It will be reflected in the line y x. It will be reflected in the line y 1.
Short Answer 7. If the range of function y f (x) is {y y 4}, state the range of the new function g(x) f (x 2) 3.
C y 2 x 3
8. As a result of the transformation of the graph of y f (x) into the graph of y 3f (x 2) 5, the point (2, 5) becomes point (x, y). Determine the value of (x, y).
D y 2 x 3
9. The graph of f (x) is stretched horizontally
B y 2 x 3
4. Which of the following transformations would produce a graph with the same x-intercepts as y f (x)? A B C D
y f (x) y f (x) y f (x 1) y f (x) 1
1 about the y-axis and then 2 1 stretched vertically by a factor of about 3
by a factor of
the x-axis. Determine the equation of the transformed function.
10. A function f (x) x2 x 2 is multiplied by a constant value k to create a new function g(x) k f (x). If the graph of y g(x) passes through the point (3, 14), state the value of k.
Extended Response 11. Sketch the graph of the inverse relation. a)
b)
12. The graphs of y f (x) and y g(x) are shown. a) If the point (1, 1) on y f (x) maps onto the point (1, 2) on y g (x), describe the transformation and state the equation of g (x). b) If the point (4, 2) on y f (x) maps onto the point (1, 2) on y g (x), describe the transformation and state the equation of g (x).
13.Consider the graph of the function y f (x). a) Describe the transformation of y f (x) to y 3f (2 (x 1)) 4. b) Sketch the graph.
14. A function is defined by f (x) (x 2)(x 3). a) If g(x) kf (x), describe how k affects the y-intercept of the graph of the function y g(x) compared to y f (x). b) If h(x) f (mx), describe how m affects the x-intercepts of the graph of the function y h(x) compared to y f (x).
15. Complete the following for the quadratic function f (x) x2 2x 1. a) Write the equation of f(x) in the form y a(x h)2 k. b) Determine the coordinates of the vertex of x f ( y). c) State the equation of the inverse. d) Restrict the domain of y f (x) so that its inverse is a function.
Chapter 1 Review - Answers 1. D 2. C 3. A 4. A 5. B 6. A 7. { y | y 1, y R} 8. (0, 20) 9. y
1 f (2 x) 3
10. k 3.5 11. a)
b)
12. a) vertical stretch by a factor of 2 about the x-axis; g ( x) 2 x b) horizontal stretch by a factor of
1 about the 4
y-axis; g ( x) 4 x 13. a) vertical stretch by a factor of 3 about x-axis, horizontal stretch by a factor of the y-axis, horizontal translation 1 unit right, vertical translation 4 units up b)
14. a) y-intercept 6k; The original y-intercept is multiplied by the value of k. b) x-intercept =
2 3 m
,
m
; The original x-intercept is multiplied by the value of
15. a) y (x 1)2 b) (0, 1) c) y 1 x d) x 1 or x 1
1 . m
1 about the y-axis, reflection in 2
Chapter 2 (Radical Functions) Review What is the graph of y
Multiple Choice For #1 to #5, choose the best answer.
A
1. Which radical function has a domain of {x | x 2, x R} and range of { y | y 3, y R} ? A y 3 x 2 B y 3 x2 C y 3 x 2 B
D y 3 x2 2. Given that the point (x, 4x2), x 0, is on the function y f (x), which of the following is the point y f ( x) on? A B C D
( x , 4x2) (x, 2x) (x, 2x2) ( x , 2x)
3. The radical function y f ( x) has an x-intercept at 2. If the graph of the function is stretched horizontally by a factor of
1 2
C
about
the y-axis, what is the new x-intercept? A 2 C
1 2
B 1 D
1 4
4. This graph is of the function y f (x). What is the graph of y A
f ( x) ?
D
f ( x) ?
5. The point (4, 10) is on the graph of the function f ( x) k 3( x 1) 4. What is the value of k? A 2 B 2 C 2
D
1 2
Short Answer 6. The point (4, y) is on the graph of f ( x) x . The graph is transformed into g (x) by a horizontal stretch by a factor of 2, a reflection about the x-axis, and a translation up 3 units. Determine the coordinates of the corresponding point on the graph of g (x). 7. State the invariant point(s) when y x 25 2
is transformed into y x 2 25. 8. The graph of is horizontally translated 6 units left. State the equation of the translated function g (x). Extended Response 9. This graph is of the function y f (x).
10. a) Describe the transformation of y x to y 4 2 x 3. b) State the domain and range of the transformed function. c) Explain how the graph of the transformed function can be used to solve the equation 0 2 x 3 4. 11. The graph of f ( x) x is stretched vertically by a factor of 4, reflected in the y-axis, vertically translated up 3 units, and horizontally translated left 5 units. Write the equation of the transformed function, g (x), and sketch the graph. 12. What real number(s) is exactly one third its square root? 13. Mary solved the radical equation x 1 3x 7 algebraically and determined that the solution is x 3 and x 2. John solved the same equation graphically. He sketched graphs of the functions y x 1 and and determined that the point of intersection is (3, 4). a) Determine the correct solution to the equation x 1 3x 7. b) Explain how Mary’s and John’s solutions relate to the correct solution. 14. a) Solve 3x 1 2 x 2 2. b) Identify any restrictions on the variable. c) Verify your solution.
a) Determine the equation of the graph in the form f ( x) b( x h) k . b) Determine the equation in simplest form.
15. On a clear day, the distance to the horizon, d, in kilometres, is given by d 12.7h , where h is the height above ground, in metres, from which the horizon is viewed. If you can see a distance of 32.5 km from the roof of a building, how tall is the building, to the nearest tenth of a metre?
Chapter 2 – Review – Answers 1. A 2. B 3. B 4. D 5. B 6. (8, y 3) or (8, 1) 7. (5, 0), (5, 0) 8. g ( x) 2( x 6)
9. a) f ( x) 4( x 1) 2 b) g ( x) 2 x 1 2 10. a) vertical stretch by a factor of 2 about the x-axis, translation down 4 units, translation right 3 units b) domain: {x | x 3, x R} ; range: { y | y 4, y R} c) The solutions to 0 2 x 3 4 are the x-intercepts of the graph of y 2 x 3 4. 11. g ( x) 4 ( x 5) 3
12.
1 9
13. a) x 3 b) Example: Since Mary used an algebraic method, she must verify her answers. Only x 3 is a solution. John determined the point of intersection, but only the x-coordinate of the point of intersection is the solution. 1 14. a) x , x 1 b) There are no restrictions on the variable. 7 15. 83.2 m
Chapter 3 Polynomial Functions Review Multiple Choice For #1 to # 4, choose the best answer. 1. The partial graph of a third-degree polynomial function of the form P(x) ax3 bx2 cx d is shown.
Which statement about the values of a and d is correct? A a 0 and d 0 B a 0 and d 0 C a 0 and d 0 D a 0 and d 0
2. Which polynomial function has zeros of 3, 1, and 2, and y-intercept 6? A (x 3)(x 1)2(x 2) B (x 3)(x 1)(x 2) C (x 3)(x 1)(x 2) D (x 3)(x 1)(x 2)2 3. The partial graph of the function P(x) ax4 bx3 cx2 dx e is shown. Consider the following statements. i) The y-intercept at point S is equal to the constant e. ii) a 0 iii) The multiplicity of the zero at point T is 2. A Only statement i) is true. B Only statement ii) is true. C Only statement iii) is true. D All three statements are true.
4. The graph of the function f (x) (x 4)(x 2)(x 6) is transformed by a horizontal stretch by a factor of 2. Which of these statements is true? A The new zeros of the function are 12, 8, 4. B The new zeros of the function are 3, 2, 1. C The new y-intercept is 96. D The new y-intercept is 24. Short Answer 5. When f (x) x3 7x2 kx 17 is divided by x 5, the remainder is 2. Determine the value of k. 6. The partial graph of the third-degree polynomial function P(x) a(x b)(x c)(x d) is shown. Determine the value of a.
7. If P(x) x4 bx2 c, P(1) 9, and P(3) 25, what are the values of b and c? 8. The volume of a box is represented by the function V(x) x3 6x2 11x 6. The height of the box is x 2. If the area of the base is 24 cm2, determine the height of the box. 9. Determine the largest possible solution to the polynomial equation x3 10x2 33x 36. 10. Perform the division (x3 5x2 x 5) ÷ (x 2). Express the result in the form
P( x) xa
Q( x )
R xa
.
11. Factor x4 13x2 12x completely. 12. The graph of y x3 x2 cx 4 has an x-intercept of 1. Determine the value of c and the remaining x-intercepts.
13. Graph the function f (x) x3 x2 10x 8. State the x-intercepts, y-intercepts, and the zeros of the function. Determine the intervals where the function is positive and the intervals where the function is negative. 14. The graph of the function f (x) x3 is translated horizontally to create g(x). If the point (4, 8) is on g(x), determine the equation of g(x). 15. The function f (x) x4 is horizontally stretched by a factor of
1 2
about the y-axis, reflected in the x-
axis, and translated vertically 1 unit up. Explain how the domain and range of f (x) are changed by the transformation. Chapter 3 – Review – Answers 1. C 2. A 3. D 4. A 5. k 7 6. a
1 2
7. b 8, c 16 8. 5 cm 9. x 4 10.
x3 5 x 2 x 5 x2
x 2 3x 7
9 x2
11. x(x 1)(x 3)(x 4) 12. c 4; x-intercepts: 2, 2 13.
x-intercepts: 4, 1, 2; y-intercept: 9; zeros: 4, 1, 2; positive intervals: (4, 1), (2, ); negative intervals: (,4), (1, 2) 14. g(x) (x 2)3 15. The domain {x | x R} does not change under this transformation. The range changes due to the reflection and the translation; it changes from { y | y 0, y R} to { y | y 1, y R}.
Chapter 4 Trig Function / Unit Circle Review Multiple Choice For #1 to #5, choose the best answer. 1. What is the exact value of csc 2 2
B
2 2
C 2
D
2
A
2. Determine tan if sin
7 ? 4
12 and 13
cos 0. A
12 5
B
5 12
C
5 12
D
12 5
3. What are the coordinates of P
if P() is the point at the intersection of the terminal arm of angle 7 6
and the unit circle?
3 1 A , 2 2
1 3 B , 2 2
3 1 , C 2 2
1 3 D , 2 2
4.Suppose tan2 tan 0 and 0 2. What does equal? A
5 , 4 4
C
0, , ,
4
5 4
B
3 7 , 4 4
D
0,
3 7 , , 4 4
5. What is the general solution of the equation 2 cos 1 0 in degrees? A 240 360n, 300 360n, n I B 60 360n, 300 360n, n I C 60 360n, 120 360n, n I D 120 360n, 240 360n, n I Short Answer 6. Convert to radian measure. State the method you used to arrive at your solution. Use each conversion method at least once. Give answers as both exact and approximate measures to the nearest hundredth of a unit. a) 270 b) –540 c) 150
d) 240
7. Convert the following radian measures to degree measure. State the method you used to arrive at your solution. Use each conversion method at least once. Give answers as approximate measures to the nearest hundredth of a unit. a) 3.25
b) 0.40
c)
7 4
d)
–5.35
9. Use the information in each diagram to determine the value of the variable. Give your answers to the nearest hundredth of a unit. c) a)
b)
d)
10. Determine the exact value of sin 2
2 cos (120) tan . 5 6
7 4
11. Given that sin 0.3 and cos 0.5, determine the value of tan to the nearest tenth. 12. If sin
3 , determine all possible coordinates of P() where the terminal arm of intersects the 2
unit circle.
3 1 13. If P() = , , what are the coordinates of P ? 2 2 2
Extended Response 14. Consider an angle of
4 radians. 5
a) Draw the angle in standard position. b) Write a statement defining all angles that are coterminal with this angle.
15. The point (3a, 4a) is on the terminal arm of an angle in standard position. State the exact value of the six trigonometric ratios. 16. Solve the equation sec2 2 0, . 17. Consider the following trigonometric equations. A 2 sin 3 0
B
2 cos 1 0
C
2 2 sin cos 2 sin 6 cos 3 0
a) Solve equations A and B over the domain 0 . b) Explain how you can use equations A and B to solve equation C, 0 . Chapter 4 – Review – Answers 1. C 2. A 3. A 4. C
5. D
3 ; 4.71 6. a) Example: unitary method; 2
b) Example: proportion method; 3; 9.42 5 ; 2.62 6 4 ; 4.19 d) Example: unitary method; 3
c) Example: unit analysis;
7. a) Example: proportion method; 186.21° b) Example: unitary method; 22.92° c) Example: unit analysis; 315° d) Example proportion method; 306.53° 8.
9 2
9. a) 133.69 or 2.33 b) a 31.85 cm 10.
c) r 6.99 m d) a 4.28 ft
3 4
11. 0.6 1 3 1 3 12. , , , 2 2 2 2
1 3 13. , 2 2 14. a)
4 2n, n I 5 15. sin 4 , cos 3 , tan 4 , csc 5 , sec 5 , cot 3 5 3 5 3 4 4 16. , 4 4 2 17. a) Equation A: , ; Equation B: 3 3 4
b)
b) Equation C is the product of Equation A times Equation B (i.e., AB C). Therefore, the solution to Equation C is the solutions to A and B:
2 , , . 4 3 3
Chapter 5 Trigonometric Graphs Review Multiple Choice For 1 to 4, select the best answer. 1. The minimum value of the function f () a cos b( c) d, where a 0, can be expressed as A ad Badc C d |a|
D
d a b
2. Which of the following is the equation of the sine wave graphed below?
A y 8 sin x 4
B y 8 sin x 2
C y 8 sin (2x)
D y 8 sin (4x)
1
1
3. When the graph of y sin has been transformed according to the directions 1 y sin x , the horizontal phase shift of 6 2
the resultant graph is units to the right 12 B units to the left 2 C units to the right 2
A
D 3 units to the left
4.Colin is investigating the effect of changing the values of the parameters a, b, c, and d in the equation y a sin b( c) d. He graphed the function f (x) sin . He then determined that the transformation that does not change the x-intercepts is described by A g () 2 sin B h () sin 2 C r () sin ( 2) D s () sin 2 Short Answer 5. The pedals on a bicycle have a maximum height of 30 cm above the ground and a minimum height of 8 cm above the ground. Out for a ride, a cyclist pedals at a constant rate of 20 cycles per minute. Write an equation for this periodic function in the form y a sin (bt) d. 6. Write the equation of a cosine function in the form y a cos b(x c) d, with an amplitude of 2, period of 6, phase shift of units to the left, and translated 3 units down. 7. State the amplitude and range for the graph of y 5 sin 3. 8. a) What system of equations can be solved using the graph below? b) State one single equation that can be solved using the graph. Then, give the general solution to the equation.
9. Consider the graph of y tan , where is measured in radians. a) What is the general equation of the asymptotes of the graph? b) What are the domain and range of the graph of the function? 10. A boat is travelling along a narrow river between two observers, as shown. The driver and both observers can hear the boat’s motor, but the sound that each of them hears is different, depending on their location in relation to the boat. The observer in front of the boat hears a higher-pitched noise than the driver hears. The observer behind the boat hears a lower-pitched sound than the driver hears.
a) Sketch and label a graph of the bottle’s distance below the pier for 15 s. Assume that at t 0, the bottle is closest to the bottom of the pier. b) Determine the period and the amplitude of the function. c) Which function would you consider to be a better model of the situation, sine or cosine? Explain. d) Write the equation of the sine function that models the bottle’s distance below the pier. e) You can reach 0.9 m below the pier. Use your equation to estimate the length of time, to the nearest tenth of a second, that the bottle is within your reach during one cycle. f ) Write the cosine function for this situation. Would your answer for part e) change using this equation? Explain. 12. Two sinusoidal functions are shown in the graph.
a) Suppose the sound of the boat is modelled by a sinusoidal function. Which characteristic—amplitude, period, or range—varies among the three sound waves? b) Which parameter in the equation y a sin bt d would change if all three functions were graphed? c) Which observer’s model equation would have the largest value of the changing parameter? Extended Response 11. You are sitting on a pier when you notice a bottle bobbing in the waves. The bottle reaches 0.8 m below the pier, before lowering to 1.4 m below the pier. The bottle reaches its highest point every 5 s.
a) Which characteristics of the two graphs are the same? b) Which parameters must change to transform the graph of f (x) to the graph of g(x)? c) Determine the equation for each of the graphs in the form y a cos b(x c) d.
Chapter 5 – Review – Answers 1. C 2. D 3. D 4. A 2 t 19 3 1 6. y 2 cos ( x ) 3 3
5. y 11sin
7. amplitude: 5; range: {y | –8 y 2, y R} 8. a) y 2cos x and y 1 b) 2cos x 1; x 60° 360n, n I, and x 300° 360n, n I n n I 2 b) domain:{ | n R, n I} 2
9. a) x
range: { y | y R} 10. a) period b) b c) Observer B 11. a)
b) amplitude is 0.3 m, period is 5 s c) Ensure that answers are accompanied by an explanation. Example: Cosine curve may not have a phase shift if you consider a negative a value (that is, a reflection in the x-axis). 2 5 d) d 0.3 sin t 1.1 5 4 e) 1.4 s f ) d 0.3 cos
2 t 1.1; Both equations model the same graph, so the result of the calculation would be the same. 5
12. a) amplitude, horizontal phase shift b) period or b value, and horizontal central axis or d value c) f ( x) 6 cos 3x 6, g ( x) 6 cos x 2
Chapter 6 Trig Equations and Identities Review Multiple Choice For 1 to 5, choose the best answer. cot 2 . 1 cot 2
1. Simplify the expression A cos2
B sin2
C
tan2
D sec2
C
1
D 2
C
cos2
D sin2
C
2
D not possible
B
1 cos2 cos2 tan2
D
cos2
2. The value of (sin x cos x)2 sin 2x is A 1
B 0
3. The expression
1 tan 2 is equivalent to 1 tan 2
A cos 2
B sin 2
4. If you simplify sin ( x) sin ( x) it is A 2
B 0
5. Which of the following is not an identity? A sec cos sin tan C csc cos tan
cos tan
1 cos2 2
Short Answer
5π 6. Determine the exact value of sin . 12 7. Given
sin 2 x 1.23. What is the value of cos x? 1 cos x
8. If 5 7 sin 2 cos2 0 on the domain 90 180, what is the value of ? 9. If cosθ
3π 5 π , πθ , determine the exact value of sin θ . 2 13 2
y y 10. What single trigonometric function is equivalent to sin (3 y) cos cos(3 y)sin ? 2
2
Extended Response π 11. Consider the equation sin x csc x 1 2
a) Verify the equation is true for x
π . 2
b)
Is the equation an identity? Explain.
12. Consider the equation sin2 x cos4 x cos2 x sin4 x. a) Verify the equation for x 30.
b)
Prove the equation is an identity.
13. Consider the equation
tan x sec x sin x . cot x 1 sin x
a) State the non-permissible values on the domain 0 x 360. b) Prove the equation is an identity algebraically. 14. Solve sin 2x cos x 0 algebraically for the domain x . 15. Solve csc2 x 4 cot2 x algebraically. State the general solution in radians.
Chapter 6 – Review – Answers 1. A 9.
2. C
3. A
4. B
5. D
6.
6
2
4
8. 150
10. sin 2 5y
5 13
Left side sin x 2
Right side csc x 1 csc 1 2 11 0
sin 2 2
11. a)
sin
b) No; it is not true for all permissible values of x.
0 Left side sin 2 x cos 4 x sin 2 30 cos 4 30 2 3 1 2 2
12. a)
4
13 16
13. a) x 0, 90, 180, 270, 360 b) Example:
5 , 2 6 6
14. ,
15.
7. 0.23
2 n, n; n I 3 3
Right side cos 2 x sin 4 x cos 2 30 sin 4 30 2
4 3 1 2 2
13 16
Chapter 7 Exponential Functions Review Multiple Choice For #1 to #6, choose the best answer. 1. What is the y-intercept for the graph of y bx 2, b > 1? 1 b2 1 C 2 b
A
6. Which function(s) would you graph to solve the equation 16
b2
B y1 16
D
2
C y1
5. Mary was asked to solve for x and y in the exponential equations 5x 3y 1 and 1 5
25x y . Which of the following linear equations would lead to a correct solution? A x 3y 1, x y 1 B x 3y 0, 2(x y) 1 C x 3y 1, 2x y 1 D x 3y 0, x y 2
1 2
1 2x
4x 3
1 2
1 2
4x 3
graphically.
4x 3
1
16 2
D y1 4x, y2
1 2
x
4x 3
Short Answer 7. Given the function f(x) 2x, match the graph with the correction equation. a) y f(x) b) y f(x) c) y f 1(x) d) y f(x) I
3. The graph of f(x) ax, a > 1, is transformed into g(x) 4ax 3 2. Which characteristic remains the same? A domain B range C x-intercept D y-intercept 4. The graph of the function f(x) 3ax 2, a > 0, has the same horizontal asymptote as which of the following? A y f(x) 4 B y f(x) 2 C y f(x) 2 D y f(x) 4
A y1 160.5x, y2 0.54x 3
B
2. In the equation y bx, b > 1, x is replaced by x 3 and y is replaced by y 4. Which of the following statements describes the transformation? A The point (x, y) on the graph of y bx has been transformed to the point (x 3, y 4). B The point (x, y) on the graph of y bx has been transformed to the point (x 3, y 4). C The graph of y bx has been translated 4 units to the right and 3 units up. D The graph of y bx has been translated 3 units to the left and 4 units down.
1 x 2
II
III
IV
8. The function f(x) 5(2x) is transformed by a translation 2 units right and 5 units down. The transformed function passes through the point (x, 10). Determine the value of x. 9. What vertical translation would be applied to y 4(3x) so that the translation image passes through (2, 37)? 10.
Solve for x. x
a) 3 2 81 3
b)
9 16
x 2
64 27
x
Extended Response 11.
You are given the functions y 2x and y 2(2x) 3. a) Sketch the graphs of the functions on the same grid. b) Describe the transformation from y 2x to y 2(2x) 3. c) State the range and the equation of the horizontal asymptote for each function. d) Determine the value of y when x 400 for each function. Explain how these results relate to your answers to part c).
12.
Consider the graph of the functions f and g. a) Determine the equation of the transformed function g(x). b) Describe the transformation of f(x) to g(x). c) Use the graphs to solve the equation f(x) g(x), to the nearest hundredth.
13.
14.
A single cell of the bacterium E. coli would, under ideal circumstances, divide every 20 minutes. a) If a culture begins with 1 bacterium, write the equation for the number of bacteria after n minutes. b) Determine, to the nearest minute, the time it takes for the culture to grow to 1024 bacteria. c) If each bacterium has a mass of roughly 1012 g, what is the mass of the bacteria after 1 day, to the nearest kg? A town had a population of 2200 people in 1990. Each year the population has decreased by 10%. a) Write an equation to represent the population of the town. b) What will the population be in the year 2020? c) When will the population be less than 50 people?
Chapter 7 – Review – Answers 1. C 2. A 3. A 4. D 7. a) III b) I c) II d) IV 8. x = 2 9. vertical translation up 1 unit 10. a) 9 b) 4 11. a)
5. B
6. A
b) vertical stretch by a factor of 2 about the x-axis, and a vertical translation down 3 c) y 2x: range is y 0, horizontal asymptote is y 0; y 2(2x) 3: range is y 3, horizontal asymptote is y 3 d) When x 400, y 2400 0 and y 2(2400) 3 3, both of which correspond to each function's horizontal asympote. These values of x are so large that the y-values are extremely close to the same value as the horizontal asymptote. However, the calculator rounds off the value. 12. a) g(x) 2x 4 2 b) horizontal translation right 4, vertical translation up 2 c) x 1.09
n
13. a) A 1 2 20 , where A is the number of bacteria, and n is the time, in minutes. b) 200 min c) 4 722 366 kg 14. a) P 2200(0.9)n, where n years since 1990 and P population b) 93 c) 36 years after 1990: 2026
Chapter 8 Logarithmic Functions Review Multiple Choice For #1 to 6, select the best answer. 1. The graph of f (x) logb x, b > 1, is translated such that the equation of the new graph is expressed as y 2 f (x 1). The domain of the new function is A {x | x > 0, x R} B {x | x > 1, x R} C {x | x > 2, x R} D {x | x > 3, x R} 2. The x-intercept of the function f (x) log5 x 3 is A 53 B 0 C 1 3. The equation y
1 3
y 23
B
53
D
23y x
log 2 x can also be written as y
x
A
D
x 23
C 23x y
4. The range of the inverse function, f 1, of f (x) log4 x, is A { y | y > 0, y R} B { y | y < 0, y R}
C { y | y ≥ 0, y R}
D { y | y R}
5. A graph of the function y log3 x is transformed. The image of the point (3, 1) is (6, 3). The equation of the transformed function is A y 3 log3 (x 3) B y 3 log3 (x 3) C y 3 log3 (x 3) D y 3 log3 (x 3) 6. If log27 x y, then log9 x equals A
3y 2
B
2y
C
3
3y
D
4y
Short Answer 7. If log3 5 x, express 2log3 45
1 2
log 3 225 in terms of x.
8. Determine the value of x algebraically. a) log4 x 3 d) log3 (x 1)2 2
b) log x 64 e)
2 3
c)
5log5 25 x
log2 (logx 256) 3
9. Solve for x. a) log (2x 3) log (x 2) log (2x 1) b) log (x 7) log (x 3) log (2x 1) c) 2 log2 (x 4) log2 x 1 10.
The point (6, 4) lies on the graph of y logb x. Determine the value of b to the nearest tenth.
Extended Response 11.
Solve the equation 5x 104, graphically and algebraically. Round your answer to the nearest hundredth.
12.
Given f (x) log3 x and g(x) log3 9x. a) Describe the transformation of f (x) required to obtain g(x) as a stretch. b) Describe the transformation of f (x) required to obtain g(x) as a translation. c) Determine the x-intercept of f (x). How can the x-intercept of g(x) be determined using your answer to parts a) or b)?
13.
Explain how the graph of y
log 4 (3x 1) 2
1 can be generated by transforming the graph of
y log4 x. 14.
Identify the following characteristics of the graph of the function y 2 log4 (x 1) 3 a) the equation of the asymptote b) the domain and range c) the x-intercept and the y-intercept
15.
An investment of $2000 pays interest at a rate of 3.5% per year. Determine the number of months required for the investment to grow to at least $3000 if interest is compounded monthly. Radioactive iodine-131 has a half-life of 8.1 days. How long does it take for the level of radiation to reduce to 1% of the original level? Express your answer to the nearest tenth.
16.
Chapter 8 – Review – Solutions 1. 2. 3. 4. 5. 6. 7.
B A D A A A x3
8. a)
1 64
b) 512 c) 25 d) 4, 2 e) 2
9. a) 3.5 b) no solution c) 8 10. 0.6 11. 2.89 12. a) horizontal stretch by a factor of
1 9
about the y-axis
b) vertical translation 2 units up c) x-intercept of f (x) is 1; the x-intercept of g(x) is 13. vertical stretch by a factor of horizontal translation
1 3
1 2
1 9
, since g(x) is a result of a horizontal stretch by a factor of
about the x-axis, a horizontal stretch by a factor of
units right, and a vertical translation 1 unit up 7
14. a) x 1 b) domain: {x x 1, x R}; range: { y y R} c) x , y 3 8
15. 140 months 16. 53.8 days
1 3
1 9
about the y-axis, a
Chapter 9 Radical Functions Review 5. Which of the following is true of the
Multiple Choice For #1 to #8, choose the best answer. 1. The x-intercept of y is the value of k? A 1.0 C 2.5
k 2 is 0.5. What x 1
2x . Which 1 x2
statement is false? A g(x) has two vertical asymptotes. B g(x) is not defined when x 0. C g(x) has one zero. D g(x) is a rational function. 3. Consider the functions f(x) x x2, g(x) 2x 1, and h( x)
f ( x) . Which statement is true? g ( x)
A f(x), g(x), and h(x) have the same domain. B The zero of f(x) is the vertical asymptote of h(x). C The non-permissible value of h(x) is the zero of g(x). D h(x) is equivalent to y 0.5x 0.25. 4. Consider the following graph of the function
f ( x)
2x 1 . xr
3 6? x2
A It has a zero at x 2. B Its range is {yy R}. C It is equivalent to y
B 1.5 D 3.0
2. Consider the function g ( x)
rational function y
6x 9 . x2
D It has a vertical asymptote at x 6. 6. The graph of which function has a point of discontinuity at x 1? A y C y
x 1 x 1 2
x2 1 x 1
B y
x 1 x2 1
D y
x2 1 x 1
7. Which function has a domain of {xx 1, x R} and a range of {yy 3, y R}? A y
x 3 x 1
B y
3x 2 3x x2 4 x 3
C y
3x x 1
D y
3x2 x2 x
8. How many roots does the equation 8 1 have? 1 x 2 16 x4
A 0 C 2
B 1 D 3
Short Answer 9. a) Sketch the graph of the function
y
x2 . x2 4
b) Identify the domain, range, and asymptotes of the function. c) Explain the behaviour of the function as the value of |x| becomes very large. 10. a) Sketch the graph of the function
y
2 1. x5
b) State the values of the x-intercept and y-intercept. What is the value of r? A 3 B 2 C 2 D 3
c) Solve 0
2 1 algebraically. x5
d) How is your answer to part c) related to your answers to parts a) and b)?
11. Select the graph that matches the given function. a) y A
2 5 ( x 3)2
b) y
2 2 3 c) y 3 2 ( x 5) ( x 5)2
B
C
Chapter 10 Function Operations Review Multiple Choice For #1 to #5, select the best answer. 1. From the graph, what is the value of (f g)(2)?
Short Answer 6. Given f (x) x and g(x) 4 x, match the combined function in set A with the graph in set B. Set A i) ( f g)(x) ii) ( f g)(x) iii) f(x)g(x) Set B A.
A 3
B0
C 2
D 4
2. Given f (x) x2 2 and g(x) x 5, which equation represents h(x) ( f g)(x)? A h(x) 2x2 5 B h(x) x2 x 3 2 C h(x) x x 5 D h(x) x2 2x 5 3. Let f (x) x 1 and g(x) x2 1. Determine
B.
f
the non-permissible values of y ( x) . g
A 1
B 1
4. If f (x)
C 1
D none
3x 1 and g(x) x2, which is the
domain of m( x)
f ( x) g ( x)
?
A {x | x 0, x R} B {x | x 0, x R}
C.
1
1
3
3
C x x , x R D x x , x R 5. Consider the functions f (x) x 2 and g(x) x 1 . Which statement is true? 2
A
f ( x) g ( x)
0, x 1
C f (x) g(x)
B f (x) g(x) 0 D (g f ) (x) 1
D.
f
iv) ( x) g
8. Given the functions f (x)
1
and g ( x)
x
1 x 1
, determine the equation of the combined function h(x).
Then state the domain of h(x). a) h(x) (f g)(x) b) h(x) (f g)(x)
c)
f
h(x) f(x)g(x)
d) h(x) ( x) g
9. Let f (x) x 1, g(x) x2 1, and h(x) 1 x. Determine each equation. a) q(x) f (x) h(x) b) p(x) g(f (x)) 10. Find two functions, f (x) and g(x), such that f (g(x)) (2x 3)2 5. Extended Response 11. Consider the functions f (x) x2 and g(x) 2x. a) Determine the equation of h( x)
f ( x)
,
g ( x)
and state the domain of h(x).
b) How does the graph of h(x) behave for large values of x? 12. Assume f (x) x and g(x) |x|. a) Determine the equation of h( x)
3 f ( x) g ( x) f ( x)
b) Sketch the graph of h(x).
.
c)
State the domain and range of h(x).
13. If f (x) x2 and h(x) x 1, then g(x) 3( f (h(x))) 5. a) Determine an equation for g(x). b) Describe g(x) as a transformation of f (x). 14. Let h(x) cos x and g(x)
1 x
. Determine the composite functions h(g(x)) and g(h(x)), the domain
Chapter 10 – Review – Answers 1. D 2. B
3. C 4. C 2x 1
8. a) h( x) d) h( x)
2
x x
x 1
5. C
6. i) D ii) A iii) B iv) C
; {x | x 0, 1; x R} b) h( x)
; {x | x 0, 1; x R}
1 2
x x
7. a) 5 b) 1 c) 1 d) 3 ; {x | x 0, 1; x R} c) h( x)
9. a) q(x) 2 b) p(x) x2 2x 2
1 2
x x
; {x | x 0, 1;}
10. f (x) x2 5; g(x) 2x 3
x
11. a) h( x ) 12. a) h( x)
x
2 x
;xR
b) as x increases, h(x) approaches 0
2 3x | x | x
b) 13. a) g(x) 3(x 1)2 5 or g(x) 3x2 6x 2 b) vertically stretched by a factor of 3 about the x-axis, translated left 1 and translation down 5
1 ; domain: {x | x 0, x R} x
14. a) h( g ( x )) cos c) domain: {x | x 0, x R}; range: {y | y 2, 4; y R}
g ( h( x )) secx ;domain: x | x
π 2
πn, n I, x R
b)
Chapter 11 Permutations and Combinations Review Multiple Choice For # 1 to #6, choose the best answer. 1. The number of 3-digit numbers, with repeats, that are multiples of 5 and less than 600 that can be formed from the digits 1, 3, 5, 7, and 9 is A 6 B 9 C 15 D 25 2. The grid shows the roads from point A to point C.
If the only allowed directions are south and east, the number of pathways from point A to point C that do not pass through point B is 9! 5!4! A 5!4! 9! 5! C 5!4! 4!
9! 5!3! B 5!4! 9! 5! D 5!4! 3!
3. The number of different arrangements of the letters of the word CONFERENCE is A
10! 3!4!
B
10! 3!
C
10! 2!3!
4. If nP3 = 2(nC4), then nC6 equals A 1716 B 3003 C 5005
D
10! 2!2!3!
D 8008
5. Given that the first 4 terms of a row in Pascal’s triangle are 1, 9, 36, and 84, which of the following is equivalent to the middle term of the next row? A 10C6 B 2(10C5) C 2(9C6) D 9C4 + 9C5 6. Which of the following statements is always correct? A The number of permutations of n different elements taken r at a time is less than the Continued top of next column >>
number of combinations of n different elements taken r at a time. B The number of permutations of n different elements taken r at a time is greater than the number of combinations of n different elements taken r at a time. C The number of permutations of n different elements taken r at a time is less than or equal to the number of combinations of n different elements taken r at a time. D The number of permutations of n different elements taken r at a time is greater than or equal to the number of combinations of n different elements taken r at a time. Short Answer 7. For a graduation ceremony, there are 8 people sitting on stage in one row. If the master of ceremonies must sit on the far left and the valedictorian must sit next to the principal, how many arrangements are possible? 8. A pizzeria advertises that it has 12 toppings. How many different pizzas can be made with 0 to 5 toppings? Assume each topping can be chosen only once per pizza. 9. Assume that a handshake takes 6 s. How long, in minutes, does it take for all 45 students in a graduating class to shake hands with each other, if everyone shakes hands once with every other person in the class? 10. Arrange the following expressions from least in value to greatest: 6! 14 P3 , 7C3, 6P4, 3 2!4! 13 C2 11. Solve for n: n C5 n C7 12. Determine the value of the term independent 5
1 of x in the expansion of 3x 2 3 . x
Extended Response 13. Mary is packing for her trip. She laid out her clothes and drew the following diagram.
a) How many different outfits are possible? b) Mary decides to add one more top and one more bottom. How many more outfits are possible? 14. Three students were selected from the 23 grade 12 students to be the president, vice-president, and secretary of the student council. Three other students from the 38 grade 10 and grade 11 students were also selected to be on the council. a) Determine the total number of ways that the 6 members of the student council can be selected. b) Explain why both permutations and combinations are needed to answer part a). 15. Solve for n algebraically: 126(nC3) = n + 1P5. 16. The social justice group is comprised of 4 grade 10 students, 5 grade 11 students, and 6 grade 12 students. Determine how many ways a committee of 5 students can be selected if a) there are no restrictions. b) there are exactly 2 grade 12 students on the committee. c) there are at most 2 grade 12 students on the committee. 17. Consider the expansion of (x + y)6. a) State the coefficients in the expansion of (x + y)6 using Pascal’s triangle. b) State the coefficients in the expansion of (x + y)6 using combinations.
c) Use the answer from part a) or part b) to expand (2x – 3y2)6. Simplify the result.
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