Trigonometric Function Graphs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Trigonometric Function Graphs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The graphs of \sin x, \cos x, and \tan x as functions of a real variable, characterized by amplitude, period, phase shift, and vertical shift.

If you track the y-coordinate of a point moving around the unit circle and plot it against the angle, you get the sine wave. It's the shape of ocean waves, sound waves, and alternating current. The general form y = a\sin(bx - c) + d lets you control four properties: how tall the wave is (a, amplitude), how fast it repeats (b, affecting period), where it starts (c, phase shift), and its vertical center (d, vertical shift).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every sinusoidal function is a transformed version of y = \sin x, controlled by four parameters that determine its shape and position.

Common stuck point: Phase shift is often confused with just c. The actual horizontal shift is \frac{c}{b}, because the b factor compresses or stretches the x-axis first.

Sense of Study hint: Write the equation in the form y = a*sin(b(x - h)) + k by factoring b out of the argument. Then read off amplitude, period, shift, and midline directly.

Worked Examples

Example 1

easy
Identify the amplitude, period, phase shift, and vertical shift of y=3\sin(2x-\pi)+1. Write in standard form y=a\sin(b(x-h))+k.

Solution

  1. 1
    Rewrite: y=3\sin(2x-\pi)+1=3\sin\!\left(2\left(x-\frac{\pi}{2}\right)\right)+1.
  2. 2
    Read off parameters: a=3 (amplitude), b=2 (period =2\pi/2=\pi), h=\pi/2 (phase shift right \pi/2), k=1 (vertical shift up 1).
  3. 3
    Summary: oscillates between k-|a|=1-3=-2 and k+|a|=1+3=4, with period \pi, starting phase-shifted to the right by \pi/2.

Answer

Amplitude =3, Period =\pi, Phase shift =\pi/2 right, Vertical shift =1; range [-2,4]
Factoring out b from the argument reveals the phase shift h=c/b. The four parameters a,b,h,k completely determine the shape and position of the sinusoidal graph.

Example 2

hard
Write the equation of a cosine function with amplitude 4, period 6, phase shift right 1, and vertical shift down 2.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
State the amplitude and period of each: (a) y=\sin(4x), (b) y=5\cos(x), (c) y=-2\sin\!\left(\frac{x}{3}\right).

Example 2

medium
A sound wave has the equation P(t)=0.002\sin(440\pi t) (pressure in Pa, time in seconds). Find the frequency (Hz) and explain the connection to the period.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

trigonometric functionsperiodic functionstransformation