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GCSE · Trigonometric and Exponential Graphs

7.8D Trigonometric and Exponential Graphs: Fluency and Exam Drill

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the amplitude and the period (in degrees) of y = 3 sin(2x).

(Total for Question 1 is 2 marks)

Write down the amplitude and the vertical shift of y = -2 cos(0.5x) + 1.

(Total for Question 2 is 2 marks)

State the exact values of y = sin x for x = 0, 90, 180, 270 and 360 degrees.

(Total for Question 3 is 5 marks)

For y = sin(x - 30) + 2, find the x-coordinate of the first maximum after x = 0 (in degrees) and the value of that maximum.

(Total for Question 4 is 2 marks)

Write down the amplitude and period (in degrees) of y = 4 sin(3x).

(Total for Question 5 is 2 marks)

Evaluate the exponential function y = 2^x at x = 0, x = 1 and x = -1.

(Total for Question 6 is 3 marks)

The graph of y = 5(0.5)^x is sketched. State the horizontal asymptote and find the y-intercept.

(Total for Question 7 is 2 marks)

Solve 2^x = 8.

(Total for Question 8 is 2 marks)

For y = -sin(x + 45) - 1, state the amplitude, period, phase shift (direction and degrees) and vertical shift.

(Total for Question 9 is 4 marks)

Solve sin x = 0.5 for 0 ≤ x < 360 degrees. Give your answers in degrees.

(Total for Question 10 is 2 marks)

Find the smallest positive solution (in degrees, 1 decimal place) of 3 cos x = 1.

(Total for Question 11 is 2 marks)

The graph of y = 2 sin(0.5x) + 1 is considered for 0 ≤ x ≤ 720 degrees.
(a) State the amplitude.
(b) State the period (in degrees).
(c) State the equation of the midline.
(d) Find all x in 0 ≤ x ≤ 720 for which y = 3.

(a)State the amplitude.(1)

(b)State the period (in degrees).(1)

(c)State the equation of the midline.(1)

(d)Find all x in 0 ≤ x ≤ 720 for which y = 3.(2)

(Total for Question 12 is 5 marks)

Solve 2^x = 4 sin x for 0 ≤ x ≤ 360 degrees. Explain your reasoning.

(Total for Question 13 is 3 marks)

The graph of y = a^x passes through the points (0, 1) and (2, 18). Find the value of a. Give your answer in exact form and to 3 significant figures.

(Total for Question 14 is 2 marks)

Mark scheme · 7.8D Trigonometric and Exponential Graphs: Fluency and Exam Drill

Question 1

B1 amplitude = 3 cao
B1 period = 360/2 = 180 degrees cao
Answer: Amplitude = 3; period = 180 degrees.

Question 2

B1 amplitude = 2 cao (ignore sign)
B1 vertical shift = +1 cao
Answer: Amplitude = 2; vertical shift = +1.

Question 3

B1 sin 0 = 0 cao
B1 sin 90 = 1 cao
B1 sin 180 = 0 cao
B1 sin 270 = -1 cao
B1 sin 360 = 0 cao
Answer: sin 0 = 0; sin 90 = 1; sin 180 = 0; sin 270 = -1; sin 360 = 0.

Question 4

B1 x-coordinate = 120 degrees cao (since sin has max at 90 and phase shift +30)
B1 maximum value = 3 cao (1 + 2)
Answer: First maximum at x = 120 degrees; maximum value = 3.

Question 5

B1 amplitude = 4 cao
B1 period = 360/3 = 120 degrees cao
Answer: Amplitude = 4; period = 120 degrees.

Question 6

B1 2⁰ = 1 cao
B1 2¹ = 2 cao
B1 2^-1 = 1/2 cao
Answer: At x=0: 1; x=1: 2; x=-1: 1/2.

Question 7

B1 horizontal asymptote y = 0 cao
B1 y-intercept = 5 cao (when x = 0)
Answer: Horizontal asymptote y = 0; y-intercept = 5.

Question 8

M1 use 8 = 2³ or use logs (recognise powers)
A1 x = 3 cao
Answer: 3

Question 9

B1 amplitude = 1 cao
B1 period = 360 degrees cao
B1 phase shift = 45 degrees left (because x + 45) cao
B1 vertical shift = -1 cao
Answer: Amplitude 1, period 360 degrees, phase shift 45 degrees left, vertical shift -1.

Question 10

B1 x = 30 degrees cao
B1 x = 150 degrees cao
Answer: x = 30 degrees or x = 150 degrees.

Question 11

M1 use cos x = 1/3 and take arccos
A1 x = arccos(1/3) = 70.5 degrees awrt
Answer: 70.5 degrees (to 1 dp).

Question 12

(a) B1 amplitude = 2 cao
(a) Answer: 2
(b) B1 period = 360/0.5 = 720 degrees cao
(b) Answer: 720 degrees
(c) B1 midline y = 1 cao
(c) Answer: y = 1
(d) M1 set 2 sin(0.5x) + 1 = 3 so sin(0.5x) = 1 and solve 0.5x = 90 + 360k
(d) A1 x = 180 degrees (only solution in 0 ≤ x ≤ 720) cao
(d) Answer: x = 180 degrees

Question 13

M1 compare ranges: 4 sin x lies between -4 and 4, 2^x > 0 and 2^x > 4 for x > 2
M1 for 0 ≤ x ≤ 2 degrees, 4 sin x ≤ 4 sin 2deg < 1 while 2^x ≥ 1, showing 2^x > 4 sin x
A1 conclude there are no solutions in 0 ≤ x ≤ 360 degrees cao
Answer: No solutions in 0 ≤ x ≤ 360 degrees.

Question 14

M1 use a² = 18 from point (2,18) (and a⁰ = 1 checks)
A1 a = √18 = 3 √2 = 4.24 to 3 s.f. cao
Answer: a = √18 = 3 √2 approx 4.24 (3 s.f.).

Trigonometric and Exponential Graphs: Fluency and Exam Drill - Worksheets, Questions and Revision