Graphs of Trigonometric Functions

Welcome, students 🌟 In this lesson, you will learn how the graphs of trigonometric functions behave, how to describe them, and how to use them in IB Mathematics: Analysis and Approaches HL. Trigonometric graphs are not just shapes on paper; they model tides, sound waves, seasonal temperature changes, and rotating motion. By the end of this lesson, you should be able to recognize the key features of $sin x$, $cos x$, and $tan x$, describe transformations such as stretches and shifts, and connect these ideas to trigonometric reasoning in geometry and trigonometry.

Learning objectives

Explain the main ideas and terminology behind graphs of trigonometric functions.
Apply IB Mathematics: Analysis and Approaches HL procedures related to trigonometric graphs.
Connect trigonometric graphs to geometry, circular measure, and modelling.
Summarize how graphing fits into the wider topic of Geometry and Trigonometry.
Use examples to justify properties such as amplitude, period, and asymptotes.

Understanding the basic trig graphs

The three most important trigonometric graphs are those of $y=\sin x$, $y=\cos x$, and $y=\tan x$. These graphs come from the unit circle and repeat regularly because angles rotate through the same pattern of values. In IB, you should know that the input $x$ is usually measured in radians unless a question clearly says otherwise.

The graph of $y=\sin x$ starts at the point $(0,0)$, rises to a maximum of $1$, falls back through $0$, reaches a minimum of $-1$, and then repeats. The graph of $y=\cos x$ starts at $(0,1)$ and follows the same wave shape, but shifted left by $\frac{\pi}{2}$. Both have amplitude $1$ and period $2\pi$.

The graph of $y=\tan x$ behaves differently. It has period $\pi$, passes through $(0,0)$, and has vertical asymptotes where $x=\frac{\pi}{2}+k\pi$ for integers $k$. This means the graph rises or falls without bound near those values. Unlike sine and cosine, tangent is not defined at the asymptotes.

A useful way to think about these graphs is motion. A point moving around a circle projects onto a horizontal or vertical line, and the resulting shadow follows a sine or cosine curve. This is why trigonometric graphs appear naturally in waves, oscillations, and periodic motion 🌊

Key features: amplitude, period, midline, and range

When graphing trigonometric functions, students, the first things to identify are the amplitude, period, midline, and range.

For a function of the form $y=a\sin(bx)+d$ or $y=a\cos(bx)+d$:

The amplitude is $|a|$.
The period is $\frac{2\pi}{|b|}$.
The midline is $y=d$.
The range is $d-|a|\le y\le d+|a|$.

For example, consider $y=3\sin(2x)-1$.

The amplitude is $3$.
The period is $\frac{2\pi}{2}=\pi$.
The midline is $y=-1$.
The range is $-4\le y\le 2$.

This means the graph oscillates between $-4$ and $2$, centered around $y=-1$. The factor $2$ inside the sine function makes the graph repeat twice as fast as $y=\sin x$.

For tangent functions of the form $y=a\tan(bx)+d$:

There is no amplitude, because tangent is unbounded.
The period is $\frac{\pi}{|b|}$.
The midline is $y=d$.

For example, $y=\tan\left(\frac{x}{2}\right)$ has period $2\pi$. That means one full branch of the graph takes twice as long to repeat compared with the standard tangent graph.

Transformations and how they change the graph

IB questions often ask you to describe or sketch transformed trigonometric graphs. The general forms are important:

$$y=a\sin(b(x-c))+d$$

$$y=a\cos(b(x-c))+d$$

$$y=a\tan(b(x-c))+d$$

Each parameter has a clear effect:

$a$ changes vertical stretch/compression and reflects the graph in the $x$-axis if $a<0$.
$b$ changes the period.
$c$ shifts the graph horizontally.
$d$ shifts the graph vertically.

For example, in $y=-2\cos\left(3\left(x-\frac{\pi}{6}\right)\right)+4$:

The graph is reflected in the $x$-axis because $a=-2$.
The amplitude is $2$.
The period is $\frac{2\pi}{3}$.
The graph shifts right by $\frac{\pi}{6}$.
The midline is $y=4$.

A common mistake is to confuse the phase shift with the period. The value $c$ shifts the graph; the value $b$ changes how quickly it repeats. They do different jobs.

Another important idea is that the sign of $a$ does not change the amplitude. Amplitude is always non-negative, so it is $|a|$, not $a$.

Working with graphs from equations and key points

A strong skill in this topic is sketching a graph from its equation without using a calculator. Start with the parent graph, then apply transformations.

For $y=2\sin\left(x-\frac{\pi}{3}\right)$:

Start with $y=\sin x$.
Shift right by $\frac{\pi}{3}$.
Stretch vertically by a factor of $2$.

To sketch accurately, mark important points. For sine and cosine, one cycle can be built from five key points spaced evenly by quarter-period intervals. For example, the standard sine graph over $0\le x\le 2\pi$ passes through

$$\left(0,0\right),\left(\frac{\pi}{2},1\right),\left(\pi,0\right),\left(\frac{3\pi}{2},-1\right),\left(2\pi,0\right).$$

If the period changes, divide the new period into four equal parts. This gives the key points for one cycle.

Example: For $y=\cos(2x)$, the period is $\pi$. The quarter-period is $\frac{\pi}{4}$. So one cycle can be sketched using the points

$$\left(0,1\right),\left(\frac{\pi}{4},0\right),\left(\frac{\pi}{2},-1\right),\left(\frac{3\pi}{4},0\right),\left(\pi,1\right).$$

For tangent, use the center point and asymptotes. The standard graph of $y=\tan x$ crosses at $x=0$ and has asymptotes at $x=\pm\frac{\pi}{2}$. One branch lies between each pair of asymptotes.

Intersections, equations, and reasoning from graphs

Graphs of trig functions also help solve equations. For example, solving $\sin x=\frac{1}{2}$ means finding where the graph of $y=\sin x$ intersects the horizontal line $y=\frac{1}{2}$.

On $0\le x<2\pi$, the solutions are

$$x=\frac{\pi}{6} \quad \text{and} \quad x=\frac{5\pi}{6}.$$

For equations such as $2\cos x-1=0$, rewrite them first:

$$2\cos x-1=0 \implies \cos x=\frac{1}{2}.$$

Then use known cosine values or graph intersections.

In HL work, you may need to solve equations in a given interval, sometimes using identities or algebraic manipulation. For instance, if you see

$$\sin^2 x=\sin x,$$

then factor:

$$\sin x(\sin x-1)=0.$$

So the solutions satisfy $\sin x=0$ or $\sin x=1$.

Graphing helps check whether your answers make sense. If a solution is outside the interval, or if a graph suggests more intersections than your algebra found, you should recheck your method. This is a useful exam habit ✅

Why this topic matters in Geometry and Trigonometry

Graphs of trig functions connect directly to the wider course. In geometry, angle measures in radians and the unit circle explain why sine and cosine repeat. In circular measure, the length of an arc is $s=r\theta$, and angle movement around a circle is linked to the oscillation of trig graphs.

Trigonometric graphs also support modelling. A Ferris wheel seat moving up and down can be described by a cosine function if the ride begins at the highest point. For example, if the seat height is modeled by $h(t)=8\cos\left(\frac{\pi}{6}t\right)+10$, then:

the maximum height is $18$,
the minimum height is $2$,
the midline is $h=10$,
the period is $12$ seconds.

This kind of model is powerful because it turns real motion into a mathematical function.

Graphs also prepare you for later topics such as trigonometric identities, inverse trigonometric functions, and calculus. For example, when you study derivatives later, you will see that the shape of trig graphs affects how rates of change behave. So this topic is not isolated; it is a bridge across the course.

Conclusion

Graphs of trigonometric functions help you understand periodic behavior, transformations, and equation solving. For students, the key skills are recognizing the basic graphs, identifying amplitude, period, midline, and asymptotes, and using transformations to sketch and analyze functions accurately. These graphs also connect closely to the unit circle, circular measure, and real-world periodic models. Mastering them will support your success in the Geometry and Trigonometry unit and in later IB Mathematics topics 📘

Study Notes

$y=\sin x$ and $y=\cos x$ have amplitude $1$ and period $2\pi$.
$y=\tan x$ has period $\pi$ and vertical asymptotes at $x=\frac{\pi}{2}+k\pi$.
For $y=a\sin(bx)+d$ and $y=a\cos(bx)+d$, amplitude is $|a|$, period is $\frac{2\pi}{|b|}$, and midline is $y=d$.
For $y=a\tan(bx)+d$, period is $\frac{\pi}{|b|}$ and there is no amplitude.
Horizontal shift comes from $x-c$; vertical shift comes from $+d$.
Reflect in the $x$-axis when $a<0$.
Use quarter-period points to sketch sine and cosine graphs accurately.
Solve trig equations by using graphs, exact values, and algebraic steps.
Trig graphs model real periodic situations like waves, sound, and rotating motion.
This topic links unit-circle ideas to geometry, trigonometric equations, and later calculus.