Trigonometric & exp. Graphs Quiz (Grade 7)

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Topic:

Score: 0 / 10

Trigonometric & exp. Graphs — Notes & Worked Examples Show / Hide

Explanation:

Both y = sin x and y = cos x oscillate between −1 and 1 with a period of 360°: sin x starts at (0°, 0), peaks at (90°, 1), crosses at (180°, 0), reaches (270°, −1), and returns to (360°, 0); cos x starts at (0°, 1) and is symmetrical about x = 180°. y = tan x has period 180° with vertical asymptotes at 90° and 270°.

Exponential graphs y = ab^x (b > 0) always pass through (0, a) because b⁰ = 1, and have the x-axis (y = 0) as a horizontal asymptote. When b > 1 the graph shows exponential growth; when 0 < b < 1 it shows exponential decay.

Example 1: Solve sin x = −0.5 for 0° ≤ x ≤ 360°.

Solution:

Base acute angle: sin⁻¹(0.5) = 30°

sin x is negative in 180° < x < 360° (by wave symmetry):

x = 180° + 30° = 210° and x = 360° − 30° = 330°

Answer: x = 210° and x = 330°

Example 2: The curve y = ab^x passes through (0, 5) and (3, 40). Find a and b.

Solution:

Substitute (0, 5): 5 = a × b⁰ ⇒ a = 5

Substitute (3, 40) into y = 5b^x: 40 = 5b³

b³ = 405 = 8 ⇒ b = ∛8 = 2

Answer: a = 5, b = 2

One Page Guide (PDF) Trigonometric & exp. Graphs Worksheets

Question 1

[1 mark]

Sana needs to draw the graph of y = 3^x for 0 ≤ x ≤ 4
She draws the graph shown on the grid.

Write down one thing Sana has done wrong.

Worked Solution & Answer

When x = 0, y = 3⁰ = 1, so the y-intercept should be (0, 1) C1

Answer: The graph should cross the y-axis at (0, 1), not (0, 0), because 3⁰ = 1

Question Paper (PDF) • Mark Scheme (PDF)

Question 2

[2 marks]

Here is a sketch of part of the graph of y = k^x where k is a positive constant.
The graph passes through the point with coordinates (-1, 1.5)

Find the value of k.

k =

Worked Solution & Answer

Substitute (–1, 1.5): 1.5 = k⁻¹ M1
⇒ 1/k = 3/2
⇒ k = 2/3. A1

Answer: k = 23

Question Paper (PDF) • Mark Scheme (PDF)

Question 3

[2 marks]

Sketch the graph of y = \tan x° for 0 ≤ x ≤ 360

Worked Solution & Answer

Asymptotes at 90° and 270° M1
passes through (0, 0), (180, 0), (360, 0) with period 180°. A1

Answer: Tangent curve with vertical asymptotes at x = 90° and x = 270° and x-intercepts at 0°, 180°, 360°

Question Paper (PDF) • Mark Scheme (PDF)

Question 4

[2 marks]

On the grid, sketch the curve with equation y = 2^x
Give the coordinates of any points of intersection with the axes.

Worked Solution & Answer

Passes through (0, 1) on the y-axis M1
approaches y = 0 as x → –∞ and increases rapidly for x > 0. A1

Answer: Exponential curve increasing above the x-axis through (0, 1)

Question Paper (PDF) • Mark Scheme (PDF)

Question 5 AQA Jun 2019 3H Q23

[2 marks]

Here is a sketch of the curve y = 2^x

On the axes above, sketch the curve y = 3^x

Worked Solution & Answer

y = 3^x also passes through (0, 1) M1
For x > 0, 3^x > 2^x, so the curve is steeper and lies above y = 2^x
For x < 0, 3^x < 2^x, so the curve is closer to the x-axis and lies below y = 2^x. A1

Answer: Exponential curve passing through (0, 1), steeper for x > 0 and below for x < 0

Question Paper (PDF) • Mark Scheme (PDF)

Question 6 AQA Nov 2024 3H Q23

[3 marks]

The distance of a particle from a point is d metres after t seconds.
@@TABULAR₀@@

Work out the values of a and b.

Worked Solution & Answer

At t = 0, d = 3 ⇒ a × b⁰ = 3 ⇒ a = 3. M2
At t = 1, d = 6 ⇒ 3 × b¹ = 6 ⇒ b = 2. A1

Answer: a = 3, b = 2

Question Paper (PDF) • Mark Scheme (PDF)

Question 7

[3 marks]

Here is the graph of y = \sin x° for -180 ≤ x ≤ 180

(a) Use the graph to find estimates for the solutions of
\sin x° = 0.3 for -180 ≤ x ≤ 180 [2 marks]

(b) Write down a value of x such that
\sin(x + 20)° = 0 for -180 ≤ x ≤ 180 [1 mark]

x =

Worked Solution & Answer

(a)

Read x at y = 0.3: x ≈ 18 and x ≈ 162 M1 A1

Answer (a): 18 and 162

(b)

x + 20 = 0 or 180 ⇒ x = –20 or 160 B1

Answer (b): x = -20 or 160

Question Paper (PDF) • Mark Scheme (PDF)

Question 8 AQA Jun 2025 1H Q14

[4 marks]

(a) Complete table of values for y = 2^x. [2 marks]
14 (b) Draw the graph of y = 2^x for x from –1 to 3. [2 marks]

(a) Complete the table of values for y = 2^x
@@TABULAR₀@@ [2 marks]

(b) Draw the graph of y = 2^x for values of x from -1 to 3 [2 marks]

Worked Solution & Answer

(a)

2⁻¹ = 1/2 = 0.5 M1
2¹ = 2
2³ = 8 A1

Answer (a): x = –1: 0.5 (or 1/2); x = 1: 2; x = 3: 8

(b)

Plot all points and connect with a smooth increasing curve M1 A1

Answer (b): Smooth curve drawn through (–1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8)

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Question 9

[4 marks]

(a) Sketch the graph of y = sin x° for 0 ≤ x ≤ 360. (2) (b) Solve 2 sin x° = –1 for 0 ≤ x ≤ 360. (2)

(a) Sketch the graph of y = \sin x° for 0 ≤ x ≤ 360 [2 marks]

(b) Solve the equation 2\sin x° = -1 for 0 ≤ x ≤ 360 [2 marks]

Worked Solution & Answer

(a)

Sine wave from (0,0) to peak (90,1), through (180,0), trough (270,–1), to (360,0) with 1 and –1 labelled on y-axis M1 A1

Answer (a): Sine wave from (0,0) to peak (90,1), through (180,0), trough (270,–1), to (360,0) with 1 and –1 labelled on y-axis

(b)

sin x° = –1/2. Since sin 30° = 1/2, in 3rd and 4th quadrants x = 180 + 30 = 210 and x = 360 – 30 = 330 M1 A1

Answer (b): 210, 330

Question Paper (PDF) • Mark Scheme (PDF)

Question 10 IGCSE Specimen 2H Q16

[4 marks]

The diagram shows part of a sketch of the curve y = \sin x°

(a) (i) the point P
(ii) the point Q [2 marks]

(b) Sketch the graph of y = \tan x for 0° ≤ x ≤ 360°
Show the coordinates of any points of intersection with the coordinate axes. [2 marks]

Worked Solution & Answer

(a)

sin x° crosses the x-axis at P(180, 0) and reaches its minimum at Q(270, –1) M1 A1

Answer (a): (i) (180, 0), (ii) (270, –1)

(b)

tan x has vertical asymptotes at x = 90° and x = 270° and crosses the axes at (0, 0), (180, 0) and (360, 0) M1 A1

Answer (b): y = tan x sketched with asymptotes at 90° and 270° and x-intercepts at (0, 0), (180, 0), (360, 0)

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Score: 0 / 10 (0 / 27 marks)

Trigonometric & exp. Graphs Worksheets Next: Rearranging (harder) →

Grade 7 Quizzes

19 Topics

7A - Surds	7B - Bounds
7C - Proportion (harder)	7D - Algebraic fractions
7E - The quadratic formula	7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs	7H - Rearranging (harder)
7J - Functions (harder)	7K - Iteration
7L - Area of any triangle	7M - The sine rule
7N - The cosine rule	7P - Congruent triangles
7Q - 3D pythagoras / trig.	7R - Histograms
7S - Conditional probability	7T - Exact trig. values (harder)
7U - Area of a segment