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7.8 Trigonometric and Exponential Graphs

EDEXCEL 1MA1 · Calculator allowed · about 110 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Without using a calculator, write down the exact value of each of the following.

(a)sin 30 degrees(1)

(b)cos 60 degrees(1)

(c)tan 45 degrees(1)

(Total for Question 1 is 3 marks)

The diagram shows the graph of y = sin x for 0 ≤ x ≤ 360.

(a)Write down the maximum value of sin x and the value of x, in the given range, at which it occurs.(2)

(b)Write down the period of the graph y = sin x.(1)

(c)Write down all the values of x in the range 0 ≤ x ≤ 360 for which sin x = 0.(1)

(Total for Question 2 is 4 marks)

The graph of y = cos x is shown for 0 ≤ x ≤ 360.

(a)Write down the coordinates of the two points where the graph of y = cos x crosses the x-axis, for 0 ≤ x ≤ 360.(1)

(b)Given that cos 50 degrees = 0.643 (3 s.f.), use the symmetry of the graph to write down the value of cos 310 degrees.(1)

(c)Given that cos 50 degrees = 0.643 (3 s.f.), write down the value of cos 230 degrees.(1)

(Total for Question 3 is 3 marks)

Given that sin 25 degrees = 0.423 (3 s.f.),

(a)use the symmetry of the sine graph to write down the value of sin 155 degrees.(1)

(b)Hence write down the two solutions of sin x = 0.423 in the range 0 ≤ x ≤ 360.(2)

(Total for Question 4 is 3 marks)

y = 2^x

(a)Complete the table of values for y = 2^x.
x: -2, -1, 0, 1, 2, 3
y: ___, ___, ___, ___, ___, ___(3)

(b)State the value of y when x = 0 for any graph of the form y = k^x, where k > 0.(1)

(Total for Question 5 is 4 marks)

The graph of y = 3^x is an exponential curve.

(a)Write down the coordinates of the point where the graph crosses the y-axis.(1)

(b)Write down the equation of the horizontal asymptote of the graph.(1)

(c)State whether y = 3^x is increasing or decreasing as x increases.(1)

(Total for Question 6 is 3 marks)

The graph of y = sin x is translated by the vector (0, 2) to give the graph of y = sin x + 2.

(a)Write down the maximum and minimum values of y = sin x + 2.(2)

(b)State the period of y = sin x + 2.(1)

(Total for Question 7 is 3 marks)

The graph of y = sin 2x is a transformation of y = sin x.

(a)Describe fully the transformation from y = sin x to y = sin 2x.(1)

(b)Write down the period of y = sin 2x.(1)

(c)Write down the number of complete cycles the graph of y = sin 2x makes for 0 ≤ x ≤ 360.(1)

(Total for Question 8 is 3 marks)

Using a calculator, solve sin x = 0.6 for 0 ≤ x ≤ 360, giving your answers to 1 decimal place.

(Total for Question 9 is 4 marks)

Each equation below is a transformation of y = sin x. Match each equation to the letter of the description that correctly describes its transformation.
A) Translation by vector (-90, 0)
B) Stretch, scale factor 4, parallel to the y-axis
C) Translation by vector (0, -3)
D) Stretch, scale factor 2, parallel to the x-axis

(a)y = sin x - 3(1)

A) Translation by vector (-90, 0)
B) Stretch, scale factor 4, parallel to the y-axis
C) Translation by vector (0, -3)
D) Stretch, scale factor 2, parallel to the x-axis

(b)y = sin(x + 90)(1)

A) Translation by vector (-90, 0)
B) Stretch, scale factor 4, parallel to the y-axis
C) Translation by vector (0, -3)
D) Stretch, scale factor 2, parallel to the x-axis

(c)y = 4 sin x(1)

A) Translation by vector (-90, 0)
B) Stretch, scale factor 4, parallel to the y-axis
C) Translation by vector (0, -3)
D) Stretch, scale factor 2, parallel to the x-axis

(d)y = sin(x/2)(1)

A) Translation by vector (-90, 0)
B) Stretch, scale factor 4, parallel to the y-axis
C) Translation by vector (0, -3)
D) Stretch, scale factor 2, parallel to the x-axis

(Total for Question 10 is 4 marks)

A colony of bacteria in a laboratory sample is modelled by P = 200 x 1.15^t, where P is the population and t is the time in hours after the sample was taken.

(a)Calculate the population after 5 hours, giving your answer to the nearest whole number.(2)

(b)Find the first whole number of hours, t, for which the population exceeds 500.(2)

(c)Give a reason why this exponential model may not be realistic for very large values of t.(1)

(Total for Question 11 is 5 marks)

The value of a car, V pounds, after n years is modelled by V = 15000 x 0.88^n.

(a)Calculate the value of the car after 4 years, to the nearest pound.(2)

(b)State the equation of the horizontal asymptote of the graph of V against n, and explain what it represents in this context.(2)

(Total for Question 12 is 4 marks)

The graph of y = cos(x - 90) is a transformation of y = cos x.

(a)Describe fully the transformation from y = cos x to y = cos(x - 90).(1)

(b)By considering this transformation, or otherwise, write cos(x - 90) in terms of sin x.(2)

(Total for Question 13 is 3 marks)

The graph of y = tan x has vertical asymptotes.

(a)Write down the equations of the two asymptotes of y = tan x for 0 < x < 360.(2)

(b)Write down the period of the graph y = tan x.(1)

(Total for Question 14 is 3 marks)

The graph of y = 2cos x - 1 is a transformation of y = cos x.

(a)Write down the maximum and minimum values of y = 2cos x - 1.(2)

(b)Write down the coordinates of the y-intercept of the graph.(1)

(c)State the period of the graph.(1)

(Total for Question 15 is 4 marks)

A curve has equation y = a x b^x, where a and b are positive constants. The curve passes through the points (0, 12) and (2, 48). Find the values of a and b, and hence write down the equation of the curve.

(Total for Question 16 is 5 marks)

Solve 2 sin x + 1 = 0 for 0 ≤ x ≤ 360, giving all solutions.

(Total for Question 17 is 4 marks)

The graph shows y = a sin(bx) for 0 ≤ x ≤ 360, where a and b are positive constants. The maximum point of the curve is at (45, 3), and the curve completes exactly 2 full cycles between x = 0 and x = 360.

(Total for Question 18 is 4 marks)

Two savings accounts are modelled by A = 500 x 1.03^t and B = 400 x 1.045^t, where t is the number of complete years the money has been invested. Find the first whole number of years, t, after which the value of account B first exceeds the value of account A.

(Total for Question 19 is 5 marks)

By using the symmetry property sin x = sin(180 - x), or otherwise, find all the solutions of sin(3x) = 0.5 in the range 0 ≤ x ≤ 360.

(Total for Question 20 is 5 marks)

Mark scheme · 7.8 Trigonometric and Exponential Graphs

Question 1

(a) B1 1/2 oe (e.g. 0.5)
(a) Answer: 1/2
(b) B1 1/2 oe (e.g. 0.5)
(b) Answer: 1/2
(c) B1 1 cao
(c) Answer: 1

Question 2

(a) B1 maximum value = 1
(a) B1 x = 90
(a) Answer: Maximum = 1 at x = 90
(b) B1 360
(b) Answer: 360 degrees
(c) B1 0, 180, 360 all stated (oe)
(c) Answer: x = 0, 180, 360

Question 3

(a) B1 (90, 0) and (270, 0) both required
(a) Answer: (90, 0) and (270, 0)
(b) B1 0.643 (using cos(360-x) = cos x)
(b) Answer: 0.643
(c) B1 -0.643 (using cos(180+x) = -cos x)
(c) Answer: -0.643

Question 4

(a) B1 0.423 (using sin(180-x) = sin x)
(a) Answer: 0.423
(b) B1 x = 25
(b) B1 x = 155
(b) Answer: x = 25, 155

Question 5

(a) B1 0.25 and 0.5 correct
(a) B1 1 and 2 correct
(a) B1 4 and 8 correct
(a) Answer: 0.25, 0.5, 1, 2, 4, 8
(b) B1 y = 1
(b) Answer: y = 1

Question 6

(a) B1 (0, 1)
(a) Answer: (0, 1)
(b) B1 y = 0
(b) Answer: y = 0
(c) B1 increasing (since base 3 > 1)
(c) Answer: Increasing

Question 7

(a) B1 maximum = 3
(a) B1 minimum = 1
(a) Answer: Maximum = 3, minimum = 1
(b) B1 360
(b) Answer: 360 degrees

Question 8

(a) B1 stretch, scale factor 1/2, parallel to the x-axis (oe)
(a) Answer: Horizontal stretch, scale factor 1/2, parallel to the x-axis
(b) B1 180
(b) Answer: 180 degrees
(c) B1 2
(c) Answer: 2

Question 9

M1 sin^-1(0.6) = 36.9 (awrt)
A1 x = 36.9 cao
M1 180 - 36.9 (ft their first solution)
A1 x = 143.1 cao
Answer: x = 36.9 and x = 143.1

Question 10

(a) B1 C
(a) Answer: C
(b) B1 A
(b) Answer: A
(c) B1 B
(c) Answer: B
(d) B1 D
(d) Answer: D

Question 11

(a) M1 200 x 1.15⁵
(a) A1 402 awrt
(a) Answer: 402
(b) M1 trial values shown, e.g. t=6 gives 462.6, t=7 gives 532.0
(b) A1 t = 7 cao
(b) Answer: t = 7
(c) B1 sensible limiting factor stated, e.g. limited space/food/resources so growth cannot continue indefinitely
(c) Answer: The population cannot grow forever because resources such as space, food or oxygen are limited.

Question 12

(a) M1 15000 x 0.88⁴
(a) A1 8995 awrt
(a) Answer: GBP 8995
(b) B1 V = 0
(b) B1 the value of the car decreases towards GBP 0 but never actually reaches it (oe)
(b) Answer: V = 0; the car's value keeps decreasing but never reaches zero.

Question 13

(a) B1 translation by vector (90, 0) (oe: 90 degrees to the right)
(a) Answer: Translation by vector (90, 0)
(b) M1 recognises shifted cosine graph matches the sine graph
(b) A1 sin x oe
(b) Answer: cos(x - 90) = sin x

Question 14

(a) B1 x = 90
(a) B1 x = 270
(a) Answer: x = 90 and x = 270
(b) B1 180
(b) Answer: 180 degrees

Question 15

(a) B1 maximum = 1
(a) B1 minimum = -3
(a) Answer: Maximum = 1, minimum = -3
(b) B1 (0, 1)
(b) Answer: (0, 1)
(c) B1 360
(c) Answer: 360 degrees

Question 16

M1 substitutes x=0 into y=a x b^x to find a
A1 a = 12
M1 substitutes (2,48) and a=12 to form 12b²=48
A1 b = 2 cao
B1 y = 12 x 2^x stated (ft)
Answer: a = 12, b = 2, y = 12 x 2^x

Question 17

M1 rearranges to sin x = -0.5
M1 finds reference angle sin^-1(0.5) = 30
A1 x = 210
A1 x = 330
Answer: x = 210 and x = 330

Question 18

B1 a = 3 (from the maximum value)
M1 period = 360 / 2 = 180
A1 b = 2
B1 checks/states y = 3 sin(2x)
Answer: a = 3, b = 2, y = 3 sin(2x)

Question 19

M1 attempts trial values of A and B at t=15
A1 A15 awrt 779, B15 awrt 774 (B < A)
M1 attempts trial values of A and B at t=16
A1 A16 awrt 802, B16 awrt 809 (B > A)
B1 t = 16 stated as conclusion
Answer: t = 16

Question 20

M1 identifies base solutions of sin u = 0.5 as u = 30 or u = 150
M1 recognises that since 0 ≤ 3x ≤ 1080, three repeats of the period must be considered
A1 at least 4 of the 6 values of u correct: 30, 150, 390, 510, 750, 870
A1 all 6 values of u correct with no extras
A1 divides by 3 correctly to give x = 10, 50, 130, 170, 250, 290 (cao, all 6, no extras)
Answer: x = 10, 50, 130, 170, 250, 290

Trigonometric and Exponential Graphs - Worksheets, Questions and Revision