Maths with Lemon

Graphs and Applications of Trigonometric Functions

IB Mathematics: Analysis and Approaches · Lessons 5.4 and 5.5 · AHL 3.7. Explore periodic graphs, transformations, equations from graphs and mathematical models of real-world motion.

Lesson 5.4 — Graphs (PDF) ↗Lesson 5.5 — Applications (PDF) ↗

The Three Parent Graphs

What you need to know

  • Exact values from the unit circle
  • Angles in degrees and radians
  • Coordinates, intercepts and asymptotes

Review Using the Unit Circle before graphing.

Key Points

The sine and cosine graphs repeat every \(2\pi\) radians. Tangent repeats every \(\pi\) radians and has vertical asymptotes.

Sine
\[y=\sin x\]

Range: \([-1,1]\). Starts at the origin.

Cosine
\[y=\cos x\]

Range: \([-1,1]\). Starts at a maximum.

Tangent
\[y=\tan x\]

Period: \(\pi\). Undefined at \(x=\frac{\pi}{2}+k\pi\).

Vocabulary

Amplitude: distance from midline to peak.
Period: horizontal length of one cycle.
Midline: central horizontal line.

Lesson 5.4, page 1: fill the exact-value table and sketch all three parent graphs.

Transformations of Sine and Cosine

What you need to know

  • Vertical stretch and reflection
  • Horizontal stretch and compression
  • Phase shift and vertical translation
  • Sketch one full cycle using five key points

Key Points

Use the transformed function to read off its graph features.

\[f(x)=a\sin(b(x-h))+k\]
Amplitude
\[|a|\]
Period
\[\frac{2\pi}{|b|}\]
Midline
\[y=k\]
Phase shift
\[h\]

For sine and cosine, maximum and minimum values are \(k+|a|\) and \(k-|a|\).

Worked example — from Lesson 5.4

Consider \(y=2\sin\left(3\left(x-\frac{\pi}{6}\right)\right)+1\).

\[\text{Amplitude}=2,\quad \text{Period}=\frac{2\pi}{3}\]
\[\text{Midline}:y=1,\quad \text{Shift}=\frac\pi6\]

Maximum = 3, minimum = −1. The five key points of one cycle are spaced \(\pi/6\) apart.

When using degrees, replace \(2\pi\) with \(360^\circ\) in the period formula.

Interactive Trigonometric Graph Explorer

Try this

  • Choose sine, cosine or tangent
  • Adjust amplitude / vertical scale, frequency, phase shift and midline
  • Observe how the period changes
  • Compare your sketch with the curve

For tangent, the vertical scale is not an amplitude: tangent has no maximum or minimum.

Explore transformations

The x-axis is in radians. Dashed vertical lines indicate tangent asymptotes.

Graphing Tangent

What you need to know

  • Locate vertical asymptotes
  • Find the tangent period
  • Identify its central crossing
  • Distinguish vertical scaling from amplitude

Key Points

\[f(x)=a\tan(b(x-h))+k\]
\[\text{Period}=\frac{\pi}{|b|}\]

The central crossing occurs at \((h,k)\). Vertical asymptotes occur where the tangent argument equals \(\frac{\pi}{2}+n\pi\).

Worked example

For

For the function \(f(x)=3\tan\left(x-\frac{\pi}{4}\right)+1\), the period is \(\pi\), and the central crossing is \(\left(\frac{\pi}{4},1\right)\).

, and the closest asymptotes are:

\[x=-\frac\pi4,\qquad x=\frac{3\pi}{4}\]
Tangent has neither an amplitude nor a maximum or minimum. Use its midline and asymptotes to sketch.

Writing an Equation from a Graph

Strategy

  • Find the maximum and minimum values
  • Calculate the midline and amplitude
  • Measure the period
  • Choose sine or cosine based on a convenient starting point
  • Check the phase shift

Practice with the ten graph sketches on page 2 of Lesson 5.5.

Key Points

\[k=\frac{y_{\max}+y_{\min}}2,\quad |a|=\frac{y_{\max}-y_{\min}}2\]
\[|b|=\frac{2\pi}{\text{period}}\]
Worked example — graph features

A curve has maximum 5, minimum −1 and period \(\pi\). It starts at its maximum when \(x=0\).

\[k=2,\quad |a|=3,\quad b=2\]

A suitable model is:

\[y=3\cos(2x)+2\]

Different but equivalent sine or cosine equations may describe the same graph.

Applications: Ferris Wheels and Paddle Wheels

From the worksheets

  • Ferris-wheel height and rotation
  • Riverboat paddle-wheel motion
  • Maximum and minimum heights
  • Times when a specified height is reached

Lesson 5.4, pages 5–6, and Lesson 5.5, pages 3–5.

Model 1 — Paddle wheel (Lesson 5.5)

The worksheet gives the height in feet, with time in seconds:

\[h(t)=8\sin\!\left(\frac\pi2(t-1)\right)+6\]
Amplitude

8 ft

Period

4 seconds

Midline

6 ft

Range

−2 to 14 ft

When does the point touch the water?

At the waterline, \(h(t)=0\). Therefore:

\[8\sin\!\left(\frac\pi2(t-1)\right)+6=0\]
\[\sin\!\left(\frac\pi2(t-1)\right)=-\frac34\]

Solve with a calculator, then select times in the requested interval.

Model 2 — Navy Pier Ferris wheel

\[h(t)=70\sin\!\left(\frac{2\pi}{7}(t-1.75)\right)+80\]

Its amplitude is 70 ft, period is 7 minutes, and heights range from 10 ft to 150 ft. Lesson 5.5 asks when the car reaches 100 ft and 0 ft; the latter is impossible in this model.

Applications: Tides and Periodic Data

Interpret the model

  • Amplitude describes variation from average water depth
  • Midline gives average depth
  • Period describes the time between matching high tides
  • Compare models before drawing conclusions

See Lesson 5.5, page 6.

Model from Lesson 5.5 — Maine coastline

\[d(t)=4.36\cos(0.499t)+8.79\]

Here \(d\) is the water depth in feet, and \(t\) is the time in hours after the first high tide.

\[\text{Amplitude}=4.36,\quad \text{Midline}=8.79\]
\[\text{Period}=\frac{2\pi}{0.499}\approx12.59\text{ h}\]

The model predicts a maximum depth of 13.15 ft and a minimum of 4.43 ft.

The worksheet also gives a different model for a new moon. Compare their amplitudes, midlines and periods to explain the differences in tidal patterns.

Extra: Worksheets and Revision

Material and references

Upload both PDFs alongside this HTML file, or update the links to their actual folders on your server.

Revision checklist

Parent graphs

Sketch sine, cosine and tangent over one period.

Transformations

Read amplitude, period, phase shift and midline.

Equation from graph

Find a model using key points.

Applications

Interpret period and amplitude with units.

Recommended order: Lesson 5.4 pages 1–4, then Lesson 5.5 page 2 and modelling problems on pages 3–6.

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