Graphs of Trigonometric Functions

students, welcome to one of the most important ideas in trigonometry 📈. Trigonometric graphs show how values from the unit circle and right-triangle ratios change as an angle changes. They appear in sound waves, tides, daylight, rotating wheels, and many other real-world patterns. In this lesson, you will learn how to read and sketch graphs of $\sin x$, $\cos x$, and $\tan x$, and how transformations such as $y=a\sin(bx+c)+d$ change their shape. By the end, you should be able to explain the key vocabulary, use graph features accurately, and connect these graphs to the wider IB Mathematics Analysis and Approaches SL topic of Geometry and Trigonometry.

Why trigonometric graphs matter

Trigonometric functions are special because they repeat in a regular pattern. This repeating behaviour is called periodicity. A periodic function returns to the same values after a fixed interval, called the period. For example, $\sin x$ and $\cos x$ each have period $2\pi$, while $\tan x$ has period $\pi$.

This matters in the real world because many situations repeat over time. The height of a point on a rotating wheel, the amount of sunlight during a day, and the movement of a pendulum can all be modelled using trigonometric graphs. 🌍

A graph can tell you several things at once:

the maximum and minimum values,
where the function crosses the axes,
how fast the pattern repeats,
and how the graph changes when shifted, stretched, or reflected.

For IB Mathematics Analysis and Approaches SL, you are expected to interpret these features clearly and use them to solve problems. students, this means you should not only memorize shapes, but also understand why they look the way they do.

The basic graphs of $\sin x$, $\cos x$, and $\tan x$

The graph of $y=\sin x$ starts at the origin because $\sin 0=0$. It rises to $1$ at $x=\frac{\pi}{2}$, returns to $0$ at $x=\pi$, falls to $-1$ at $x=\frac{3\pi}{2}$, and comes back to $0$ at $x=2\pi$. This pattern repeats forever.

The graph of $y=\cos x$ starts at $1$ because $\cos 0=1$. It falls to $0$ at $x=\frac{\pi}{2}$, reaches $-1$ at $x=\pi$, and returns to $0$ at $x=\frac{3\pi}{2}$. It also repeats every $2\pi$.

The graph of $y=\tan x$ behaves differently. It passes through the origin because $\tan 0=0$, rises steeply, and has vertical asymptotes where $\cos x=0$. These occur at $x=\frac{\pi}{2}+k\pi$, where $k$ is an integer. The graph repeats every $\pi$.

Key facts to remember:

$\sin x$ and $\cos x$ have range $[-1,1]$.
$\tan x$ has range all real numbers.
$\sin x$ and $\cos x$ are smooth wave-like curves.
$\tan x$ has separate branches with asymptotes.

A useful comparison is this: $\sin x$ and $\cos x$ are like ocean waves, while $\tan x$ is like a curve that shoots upward without bound near certain angles.

A simple example

Suppose you want the $x$-coordinates where $\sin x=0$ for $0\le x\le 2\pi$. From the graph, these are $x=0$, $x=\pi$, and $x=2\pi$. This is an example of using graph features to solve an equation.

Transformations of trigonometric graphs

Most IB questions do not use only the basic graphs. They often involve transformed graphs such as $y=a\sin(bx+c)+d$ or $y=a\cos(bx+c)+d$. Each parameter changes the graph in a specific way.

Vertical stretch and reflection: $a$

In $y=a\sin x$, the value of $a$ changes the amplitude. The amplitude is the distance from the midline to a maximum or minimum value. For $\sin x$ and $\cos x$, the amplitude is $|a|$.

If $|a|>1$, the graph is stretched vertically.
If $0<|a|<1$, the graph is compressed vertically.
If $a<0$, the graph is reflected in the $x$-axis.

For example, $y=3\cos x$ has amplitude $3$, so its maximum is $3$ and its minimum is $-3$.

Horizontal stretch and compression: $b$

The coefficient $b$ affects the period. For $y=\sin(bx)$ or $y=\cos(bx)$, the period becomes

$\frac{2\pi}{|b|}$

For $y=\tan(bx)$, the period becomes

$\frac{\pi}{|b|}.$

If $|b|>1$, the graph is horizontally compressed. If $0<|b|<1$, it is stretched. This means the wave repeats more quickly or more slowly.

Phase shift: $c$

The term $c$ shifts the graph horizontally. It is often easier to rewrite the function first. For example,

$ y=\sin(bx+c)=\sin\left(b\left(x+\frac{c}{b}\right)\right)$

This shows a shift left by $\frac{c}{b}$ if $c>0$, or right if $c<0$. Students often make mistakes here because the shift is not just $c$; it depends on $b$ too.

Vertical shift: $d$

The value $d$ moves the whole graph up or down. The midline is

$y=d.$

For example, $y=2\sin x+3$ has midline $y=3$, amplitude $2$, maximum $5$, and minimum $1$.

Example of a transformed graph

Consider

$y=-2\cos(3x)+1.$

Here:

amplitude is $2$,
period is $\frac{2\pi}{3}$,
the negative sign reflects the graph in the $x$-axis,
the midline is $y=1$.

Since $\cos 0=1$, the graph starts at

$y=-2(1)+1=-1.$

So the graph begins at $-1$, rises and falls around the midline $y=1$, and completes one full cycle in $\frac{2\pi}{3}$.

Key features: amplitude, period, midline, and intercepts

When analysing a trigonometric graph, students, you should identify the main features carefully.

Amplitude

Amplitude applies to $\sin$ and $\cos$ graphs, not to $\tan x$ in the same way. It tells the size of the oscillation. If the graph is $y=a\sin x+d$, then amplitude is $|a|$.

Period

The period is the length of one complete cycle. You can often find it by locating two consecutive identical points on the graph, such as two peaks or two crossings in the same direction.

Midline

The midline is the horizontal line halfway between the maximum and minimum values. For $y=a\sin(bx+c)+d$, the midline is $y=d$.

Intercepts

The $y$-intercept is found by setting $x=0$. The $x$-intercepts are values of $x$ where the function equals $0$. For example, if

$y=2\sin x,$

then the $x$-intercepts on $0\le x\le 2\pi$ are at

$x=0,\ \pi,\ 2\pi.$

For $y=\tan x$, the $x$-intercepts occur at

$x=k\pi,$

where $k$ is an integer.

Sketching and interpreting graphs in IB problems

A common IB skill is sketching graphs from equations and using graphs to solve equations. A good strategy is to follow these steps:

Identify the parent function: $\sin x$, $\cos x$, or $\tan x$.
Find the amplitude, period, and midline.
Determine any shifts or reflections.
Mark key points over one period.
Repeat the pattern if needed.

For example, sketch $y=\sin\left(x-\frac{\pi}{2}\right)+2$.

The graph of $\sin x$ is shifted right by $\frac{\pi}{2}$.
The midline is $y=2$.
The amplitude is $1$.
The graph oscillates between $y=1$ and $y=3$.

This kind of question appears often because it tests both algebra and graphical understanding. 📚

You may also be asked to solve trigonometric equations using graphs. For example, solving $\cos x=\frac{1}{2}$ on $0\le x\le 2\pi$ gives

$x=\frac{\pi}{3},\ \frac{5\pi}{3}.$

A graph helps you see why there are two solutions in one full cycle.

Conclusion

Graphs of trigonometric functions are a core part of Geometry and Trigonometry because they show how angles, ratios, and periodic patterns are connected. students, when you understand the basic shapes of $\sin x$, $\cos x$, and $\tan x$, and how transformations change them, you gain a powerful tool for solving equations, modelling real situations, and interpreting data. In IB Mathematics Analysis and Approaches SL, this topic supports later work with trigonometric identities, equations, and applications in geometry and measurement. Strong graph skills make the rest of trigonometry much easier to understand.

Study Notes

$\sin x$ and $\cos x$ both have period $2\pi$ and range $[-1,1]$.
$\tan x$ has period $\pi$ and vertical asymptotes where $\cos x=0$.
The amplitude of $y=a\sin x$ or $y=a\cos x$ is $|a|$.
For $y=\sin(bx)$ or $y=\cos(bx)$, the period is $\frac{2\pi}{|b|}$.
For $y=\tan(bx)$, the period is $\frac{\pi}{|b|}$.
In $y=a\sin(bx+c)+d$ or $y=a\cos(bx+c)+d$, the midline is $y=d$.
A positive $c$ shifts the graph left by $\frac{c}{b}$ when the function is written as $a\sin(bx+c)+d$.
A negative $a$ reflects the graph in the $x$-axis.
Key graph features include amplitude, period, midline, intercepts, and asymptotes.
Graphs are useful for solving equations such as $\sin x=0$ or $\cos x=\frac{1}{2}$.
Trigonometric graphs model repeating phenomena in science, engineering, and everyday life.