Trigonometric Graphs

Q: Do I need to plot points to sketch trig graphs?

No. Use the amplitude, period and shifts to mark the key points: intercepts, maximums, minimums and endpoints of the domain.

Q: How do I find the period of tan graphs?

The period of $y = \tan kx$ is $\frac{180°}{|k|}$, not $\frac{360°}{|k|}$.

Q: What do IEB trig graph questions focus on?

Relationships between graphs: intersections, inequalities and transformations, rather than plotting.

Trig graph questions in Paper 2 combine sketching with solving: you sketch two graphs, find where they meet by solving an equation, and read inequalities off the sketch. The focus is on how the graphs relate, not on plotting points.

Amplitude, period and shifts

For $y = a sin (k (x + p)) + q$ (and the same for cosine):

Amplitude is $∣ a ∣$ : half the distance between maximum and minimum.
Period is $\frac{360°}{∣ k ∣}$ ( $k \neq = 0$ ). For $tan$ , the period is $\frac{180°}{∣ k ∣}$ .
Horizontal shift is $p$ units left (right if $p$ is negative).
Vertical shift is $q$ units up.

Example. $f (x) = 3 sin 2 x$ has amplitude 3, period 180° and range $- 3 \leq y \leq 3$ .

$g (x) = cos (x - 30°)$ is $y = cos x$ moved 30° right. Its maximum is at $x = 30°$ and its $x$ -intercepts are at $x = 120°$ and $x = - 60°$ .

For a tangent graph, show the asymptotes. $y = tan x$ has asymptotes at $x = 90° + k \cdot 180°$ .

Points of intersection by solving

Read points of intersection off a sketch only roughly. For exact values, solve $f (x) = g (x)$ .

Example. Find where $f (x) = 2 sin x$ and $g (x) = cos 2 x$ meet for $0° \leq x \leq 360°$ .

$2 sin x = 1 - 2 sin^{2} x \Rightarrow 2 sin^{2} x + 2 sin x - 1 = 0$

$sin x = \frac{- 2 \pm 12}{4} = \frac{- 1 \pm 3}{2}$

$sin x = \frac{- 1 - 3}{2} \approx - 1, 37$ is impossible. $sin x = \frac{- 1 + 3}{2} \approx 0, 366$ gives $x \approx 21, 47°$ or $x \approx 158, 53°$ .

Reading inequalities from graphs

With both graphs sketched and their intersections found:

$f (x) > g (x)$ : where $f$ is above $g$
$f (x) \cdot g (x) < 0$ : where the graphs are on opposite sides of the $x$ -axis
$f (x) - g (x) = c$ : shift and compare

Example. For the graphs above, $2 sin x > cos 2 x$ for $21, 47° < x < 158, 53°$ .

State intervals clearly with the correct inequality signs, and stay inside the domain the question gives.

Key terms

Amplitude: Half the vertical distance between the maximum and minimum values.
Period: The horizontal length of one complete cycle of the graph.
Phase shift: A horizontal shift of the graph.
Asymptote (tangent graph): A vertical line where $tan$ is undefined.

Common mistakes

Using $360° \times k$ instead of $\frac{360°}{k}$ for the period.
Shifting $cos (x - 30°)$ left instead of right.
Reading intersection values roughly from a sketch when exact values are needed.
Missing endpoints of the given domain in sketches.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

For $f (x) = 3 sin 2 x$ , write down:

(a) the amplitude (1)

(b) the period (1)

Worked solution

Hidden - try the question first

Solution

(a) 3
(b) $\frac{360°}{2} = 180°$
(c) $- 3 \leq y \leq 3$

Question 2

5 marks total

routine

Given $g (x) = cos (x - 30°)$ for $- 180° \leq x \leq 180°$ .

(a) Write down the maximum value of $g$ and the $x$ -value where it occurs. (2)

(b) Determine the $x$ -intercepts of $g$ . (3)

Worked solution

Hidden - try the question first

Solution

(a) Maximum value 1
at $\frac{360°}{2} = 180°$
(b) $x - 30° = \pm 90° + k \cdot 360°$
$x = 120°$
or $x = - 60°$

Question 3

7 marks total

complex

Given $f (x) = 2 sin x$ and $g (x) = cos 2 x$ for $0° \leq x \leq 360°$ .

(a) Determine the $x$ -values of the points of intersection of $f$ and $g$ . (5)

(b) For which values of $x$ is $f (x) > g (x)$ ? (2)

Worked solution

Hidden - try the question first

Solution

(a) $2 sin x = 1 - 2 sin^{2} x$
$2 sin^{2} x + 2 sin x - 1 = 0$
$sin x = \frac{- 2 \pm 4 + 8}{4} = \frac{- 1 \pm 3}{2}$
$sin x = \frac{- 1 - 3}{2}$ has no solution because it is less than $- 1$
$sin x \approx 0, 366$ : $x \approx 21, 47°$ or $x \approx 158, 53°$
(b) $21, 47° < x$
$< 158, 53°$

Question 4

9 marks total

problem-solving

The graph of $f (x) = a cos b x$ has a maximum value of 3 at $x = 0°$ and a period of 180°.

(a) Determine $a$ and $b$ . (2)

(b) Determine the values of $x$ , $0° \leq x \leq 180°$ , for which $f (x) = - 1, 5$ . (3)

(d) $g (x) = cos (2 x + 60°)$ . Write down the period of $g$ and the $x$ -value of its first maximum for $x \geq 0°$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $2 sin x = 1 - 2 sin^{2} x$
$\frac{360°}{b} = 180° \Rightarrow b = 2$
(b) $3 cos 2 x = - 1, 5 \Rightarrow cos 2 x = - \frac{1}{2}$
$2 x = 120°$ or $2 x = 240°$
$x = 60°$ or $x = 120°$
(c) $f$ is at or below $- 1, 5$ between these points: $60° \leq x \leq 120°$
(d) Period $180°$
Maximum where $2 x + 60° = 360°$ , so $x = 150°$

FAQs

Do I need to plot points to sketch trig graphs?

No. Use the amplitude, period and shifts to mark the key points: intercepts, maximums, minimums and endpoints of the domain.

How do I find the period of tan graphs?

The period of $y = tan k x$ is $\frac{180°}{∣ k ∣}$ , not $\frac{360°}{∣ k ∣}$ .

What do IEB trig graph questions focus on?

Relationships between graphs: intersections, inequalities and transformations, rather than plotting.

Amplitude, period and shifts

Points of intersection by solving

Reading inequalities from graphs

Key terms

Common mistakes

Practice questions

FAQs

Need a tutor for this subject?