a) If the point (1, 1) on y f (x) maps onto the point (1, 2) on yg (x), describe the transformation and state the equation of g (x).
PC 12 Midterm Review
Chapter 1 Review Multiple Choice
1. yf (x) contains the point (3, 4). (3, 4) is thentransformed to (5, 5). Which of the following is a possible equation of the transformed function? A y 1 f (x 2) By 1 f (x 2)
C y 1 f (x 2) Dy 1 f (x 2)
2. yx is vertically expanded by a factor of 3 , & then horizontally translated 3 units left and vertically translated up 1 unit. Which of the following points is on the transformed function?
A (0, 0) B (1, 3) C (3, 1) D (3, 1)
3. is vert. expanded by a factor of 2, reflected vertically, and then horizontally translated left 3. Which equation represents the transformed function?
A B
C D
4. Which of the following transformations would produce a graph with the same x-intercepts a yf (x)?
A yf (x) Byf (x) C yf (x 1) D yf (x) 1 5. Given the graph of yf (x), what is the invariant point
under the transformation yf (2x)?
6. What will the transformation of the graph of yf(x) be if y is replaced with y in the equation yf(x)?
A Reflected in the x-axis. B Reflected in the y-axis. C Reflected on line yx. D Reflected on line y1. Short Answer
7. If the range of function yf (x) is {y y 4}, state the range of the new function g(x) f (x 2) 3.
8. When yf (x) is transformed to y3f (x 2) 5, the (2, 5) becomes (x, y). Determine the value of (x, y).
9 f (x) is stretched horizontally by a factor of and compressed vertically by a factor of . Determine the equation of the transformed function.
10. f (x) x2x 2 is multiplied by k to create a new
function g(x) kf (x). If the graph of yg(x) passes through the point (3, 14), state the value of k.
11. Sketch the graph of the inverse relation. a) b)
12. The graphs of yf (x) and yg(x) are shown.
b) If the point (4, 2) on yf (x) maps onto the point (1, 2) on yg (x), describe the transformation and state the equation of g (x).
13.Consider the graph of the function yf (x).
a) Describe the transformation of yf (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-int of the graph yg(x) compared to yf(x)
b) If h(x) f (mx), describe how m affects the x-int of the graph of yh(x) compared to yf(x).
15. Complete the following for f (x) x2 2x 1.
A (1, 0)
a) Write the equation in the form ya(xh)2k. b) What are the coordinates of the vertex of xf ( y). c) State the equation of the inverse. d) Restrict the domain of yf (x) so that its inverse is a function. Chapter 2 Review Multiple Choice
1. Which radical function has a domain of
& range of ?
A B
C D
2. Given that (x, 4x2), x 0, is on the function yf (x), which of the following is the point on? A ( 4x2) B (x, 2x) C (x, 2x2) D ( 2x) 3. The radical function has an x-int at 2. If the
graph of the function is stretched horizontally by a factor of
about the y-axis, what is the new x-int?
A 2 B 1 C D 4. This graph is of the function yf (x).
What is the graph of ?
A B
C D
5. The point (4, 10) is on the graph of the function What is the value of k? A 2 B 2 C 2 D
Short Answer
6. The point (4, y) is on 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 yx2 25 is transformed into
8. is horizontally translated 6 units left. State the equation of the translated function g (x). 9.This graph is of the function yf (x).
10. a) Describe the transformation of to
b) State the domain and range of
c) Explain how the graph of the transformed function can be used to solve the equation
11.
f
(
x
)=
√
x
is stretched vertically by a factor of 4, reflected in the y-axis, vert. translated up 3 units, and horizontally translated left 5 units. Write the equation of the transformed function, g (x), and sketch g (x),12. What number(s) is exactly one third its square root? 13. Mary solved algebraically and
determined that the solution is x 3 and x2. John solved the same equation graphically. He sketched graphs of the functions yx 1 and and determined that the point of intersection is (3, 4). a) Determine the correct solution to
b) Explain how Mary’s and John’s solutions relate to the correct solution.
a) Determine the
equation of the
graph in the form
14. a) Solve b) Identify any
restrictions on the variable. c) Verify your solution. 15. On a clear day, the distance to the horizon, d, in
kilometres, is given by 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 3 Review Multiple Choice
1. The graph of a third-degree polynomial function of the form P(x) ax3bx2cxd is shown.
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 graph of P(x)ax4bx3cx2dxe is shown
4. f (x) (x 4)(x 2)(x 6) is horizontally expanded by a factor of 2. Which statement is true?
A The new zeros of the function are12, 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 7x2kx 17 is divided by
x 5, the remainder is 2. Determine the value of k.
6. The graph of the polynomial function P(x) a(xb)(x c)(xd) is shown. Determine the value of a.
7. If P(x) x4bx2c, 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. 9. Determine the largest possible solution to the
polynomial equation x3 10x2 33x 36.
10. Perform the division (x3 5x2x 5) ÷ (x 2). Express the
result in the form .
11. Factor x4 13x2 12x completely.
12. The graph of yx3x2cx 4 has an x-int of 1.
Determine the value of c and the remaining x-ints
13. Graph f (x) x3x2 10x 8. State the x-ints, y-ints,
and the zeros of the function. Determine the intervals where the function is positive and 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 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.
Chapter 4 Review Multiple Choice
Consider the following statements.
i) The y-int at point S is equal to the constant e.
ii) a 0
iii) The multiplicity of the zero at point T is 2.
A
S
tatement i) is true.B
S
tatement ii) is true.C
S
tatement iii) is true.D All statements are true
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
1. What is the exact value of csc ?
A B C D
2. Determine tan if and cos 0.
A B C D
3. What are the coordinates of if P() is the point of intersection of the terminal arm & the unit circle?
A B C D
4. Solve for tan2 tan 0 and 0 2.
A B C D
5. Determine the general sol. in degrees: 2 cos 1 0 A 240 360n, 300 360n, n I
B 60 360n, 300 360n, n I C 60 360n, 120 360n, n I D120 360n, 240 360n, n I Short Answer
6. Convert to radian measure. Answers in exact values. a)270 b) –540c) 150 d) 240
7. Convert the following radian measures to degree measure. Round answers to 2 decimal places. a)3.25 b) 0.40 c) d) –5.35
8. The minute hand of an analogue clock completes one revolution in 1 h. Determine the exact value of the angle, in radians, the minute hand moves in 135 min. 9. Use the information in each diagram to determine the
value of the variable. Round to 2 decimal places. a) b)
c) d)
10. Simplify exactly:
11. Given that sin 0.3 and cos 0.5, determine the value of tan to the nearest tenth.
12. If , determine the coordinates of P() where the terminal arm of intersects the unit circle.
13. If what are the coordinates of
14. Consider an angle of radians. 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 B
C
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 5 Review Multiple Choice
1. The minimum value of the function f() a cos b(c) d, where a 0, can be expressed as
3. When y sin has been transformed according to the directions the horizontal phase shift is A units to the right B units to the left
C units to the right 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
ya 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
Ag() 2 sin B h() sin 2 Cr() 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. A cyclist pedals at a constant rate of 20 cycles per minute. Write an rquation for this periodic function in the form ya sin (bt) d.
6. Write the equation of a cosine function in the form ya cos b(xc) d, with an amplitude of 2, period of 6, phase shift of units left, and translated 3 units down. 7. State the amplitude and range for y5 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
9. Consider the graph of y tan , where is in radians. a)What is the general equation of the asymptotes?
b)What are the domain and range?
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)Suppose the sound of the boat is modelled by a sinusoidal function. Which characteristic—amplitude, period, or range—varies among the three waves? b)Which parameter in ya sin btd would change if
all three functions were graphed?
c)Which observer’s model equation would have the largest value of the changing parameter?
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)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? 12. Two sinusoidal functions are shown in the graph.
A
B
14. a) y-intercept 6k; The original y-intercept is multiplied by the value of k.
b) ; The original
x-intercept is multiplied by the value of
.15. a)y (x 1)2 b) (0, 1) c)
d)x 1 or x 1
a)What 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 ya cos b(xc) d.
Chapter 6 Review
Multiple Choice
1. Simplify the expression .
A cos2 B sin2C tan2 D sec2 2. The value of (sin x cos x)2 sin 2x is
A 1 B 0 C 1 D 2
3. The expression is equivalent to A cos 2 B sin 2 C cos2 D sin2 4. If you simplify sin (x) sin (x) it is
A 2 B 0 C 2 D not possible 5. Which of the following is not an identity?
A sec cos sin tan B 1 cos2 cos2 tan2
C D
Short Answer
6. Determine the exact value of
7. Given What is the value of cos x? 8. If 5 7 sin 2 cos2 0 on the domain 90
180, what is the value of ?
9. If , , determine the exact value of
10
. What single trigonometric function is equivalent to
?
11. Consider the equation
a) Verify the equation is true for . b) Is the equation an identity? Explain.
12. Consider the equation sin2x cos4x cos2x sin4x. a) Verify the equation for x 30.
b) Prove the equation is an identity.
13. Consider the equation .
a) State the non-permissible values if 0x 360.
b) Prove the equation is an identity algebraically
.
14. Solve sin 2x cos x 0algebraically if x. 15. Solve csc2x 4cot2x algebraically. State the general
solution in radians.
Answers
Chapter 1 Review Answers1. D 2. C 3. C 4. A 5. B 6. A
7. { y| y 1, y R} 8. (0, 20) 9. 10. k 3.5
11.a) b)
12. a) vertical exp. by a factor of 2 about the x-axis;
b) hor. comp by a factor of about they-axis;
13.a) vertical stretch by a factor of 3 about x-axis, horizontal
stretch by a factor of about the y-axis, reflection in the y-axis, horizontal translation 1 unit right, vertical translation 4 units up
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.
9. a)
9b) 10.a) vertical stretch by a factor of 2 about the x-axis, translation down 4 units, translation right 3 units
b) domain: ; range:
c) The solutions to are the x-intercepts of the
graph of 11.
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.
14. a) x x 1 b) There are no restrictions on the variable.
15. 83.2 m
Chapter 3 Review Answers
1. C 2. A 3. D 4. A 5.k 7 6.
7.b 8, c 16
8. 5 cm 9.x 4 10.
11.x(x 1)(x 3)(x 4) 12. c4; 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 does not change
under this transformation. The range changes due to the
reflection and the translation; it changes from
to
Chapter 4 Review Answers
1. C 2. A 3. A 4. C 5. D 6. a) b) -3; c)
6d) 7.a) 186.21° b) 22.92° c)315° d)306.53°
8. 9. a) 133.69 or 2.33 b)a 31.85 cm c)r 6.99 m
d)a 4.28 ft 10. 11. 0.6 12. ,
13. 14. a)
17. a) Equation A: ; Equation 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: .
Chapter 5 Review Answers
1. C 2. D 3. D 4. A 5.
6. 7. amplitude: 5; range: {y | –8 y 2, yR}
8. a) y 2cos x and y 1 b) 2cos x 1; x 60° 360n, n I, and
x 300° 360n, n I 9.a)x n I b) domain:{ |
R, n I} range: {y | y R} 10. a) period b)b
10c) Observer B 11. a)
12.
b)
15. sin , cos ,
tan csc ,
sec , cot
16.
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
d) e) 1.4 s f)
Both equations model the same graph, so the result of the
calculation would be the same. 12. a) amplitude, horizontal
phase shift b) period or b value, and horizontal central axis or d
value c)
Chapter 6 Review Answers
1. A 2. C 3. A 4. B 5. D 6.
7. 0.23 8. 150
9.
10. 11. a)
b) No; it is not true for all permissible values of x.
12. a)
b) Example:
13. a) x 0, 90, 180, 270, 360 b) Example:
