Graphing the Sine Function

Learn to graph the sine function and understand its key properties like amplitude, period, and phase.

Advanced25 minLesson
Printable worksheet

Definition

The sine function creates a smooth, wave-like curve called a sinusoidal wave. It oscillates between and and repeats every radians (360 degrees).
**Key properties of :**
PropertyValue
Amplitude (height from center to peak)
Period (one complete cycle)
DomainAll real numbers
Range
Zeros where is any integer
**Critical points in one period :**

Try it now

What is the amplitude of ?

Worked Examples

Graph for

1

Create a table of key values

Identify the five key points: → Key x-values identified

2

Calculate the y-values

, , , , → Points: , , , ,

3

Plot the points

Mark each point on the coordinate plane → Five points plotted

4

Connect with a smooth curve

Draw a smooth wave connecting all points → One complete sine wave

Common Mistakes

Confusing degrees and radians on the x-axis

Why it's wrong: The standard sine graph uses radians. radians equals 180 degrees.

Correct: Always check if your calculator and graph are in the same mode. For calculus and higher math, radians are standard.

Getting the phase shift direction wrong

Why it's wrong: In , the shift is to the RIGHT, not left. The minus inside creates opposite behavior.

Correct: shifts RIGHT by . shifts LEFT by .

Confusing amplitude with range

Why it's wrong: Amplitude is the distance from the midline to the peak, not the total height.

Correct: For , amplitude is 3, but the total height (range) is 6 (from to ).

Forgetting that the coefficient of affects period, not amplitude

Why it's wrong: In , compresses or stretches horizontally, changing the period.

Correct: Period . Larger means shorter period (faster oscillation).

Interactive Visual

Unit Circle

Degrees

0°

Radians

0

sin(θ)

0

cos(θ)

1

tan(θ)

0

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Linear Function Explorer

y = x
Slope (m)1
Y-Intercept (b)0
b
run
rise

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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Practice Problems

16 problems
Problem 1 of 16
Easy

What is the amplitude of ?

Why It Matters

The sine function is everywhere in the physical world:
  • Sound waves: Music and speech travel as sinusoidal pressure waves
  • Electricity: Alternating current (AC) follows a sine wave pattern
  • Ocean tides: The rise and fall of tides can be modeled with sine functions
  • Pendulums: A swinging pendulum traces a sine curve over time
  • Seasons: Temperature variations throughout the year follow a sinusoidal pattern
Engineers, physicists, and musicians all rely on understanding sine waves to design everything from bridges to synthesizers.

Real World Applications

Sound Waves and Music

Pure musical tones are sine waves. A tuning fork vibrates at 440 Hz (A note), creating a sine wave in air pressure.

Example:

The equation models a 440 Hz sound wave, where is time in seconds.

1Try It Yourself

A guitar string produces a note at 330 Hz.

What is the period of this sound wave in seconds?

Step 1: Write the mathematical expression

Use Period :

Electrical Engineering - AC Current

Household electricity uses alternating current that follows a sine wave pattern.

Example:

European outlets provide 230V at 50 Hz: where the peak voltage is about 325V.

2Try It Yourself

US electricity operates at 60 Hz with peak voltage of 170V.

Write the equation for US household voltage.

Step 1: Write the mathematical expression

Use where is peak voltage and is frequency:

Ocean Tides

Tidal heights follow a roughly sinusoidal pattern due to the moon's gravitational pull.

Example:

If high tide is at 6 AM with height 3 meters and low tide is at 12 PM with height 1 meter, the tide can be modeled with a sine function.

3Try It Yourself

A harbor has high tide of 4 meters at midnight and low tide of 0 meters at 6 hours later.

What is the amplitude and period of this tidal function?

Step 1: Write the mathematical expression

Amplitude = (max - min)/2, Period = time for one full cycle:

Key Takeaways

  • 1The sine function creates a smooth wave oscillating between and
  • 2Key points occur at (values: )
  • 3The period of the basic sine function is (one complete cycle)
  • 4In : = amplitude, = period, = phase shift, = vertical shift
  • 5Amplitude is the distance from the midline to the peak, not the total height

Frequently Asked Questions

On the unit circle, sine represents the y-coordinate. At angle 0 (pointing right on the x-axis), the y-coordinate is 0. This is why .
On the unit circle, sine represents the y-coordinate. At angle 0 (pointing right on the x-axis), the y-coordinate is 0. This is why .
The cosine graph is the same shape as sine, but shifted left by . In fact, . Cosine starts at its maximum (1) while sine starts at 0.
A negative amplitude reflects the graph across the x-axis. The wave that normally goes up first now goes down first. The peaks and troughs are inverted.

Glossary

Amplitude
The distance from the midline to the maximum (or minimum) of the wave. For , amplitude is .
Period
The horizontal length of one complete cycle. For , period is .
Phase shift
A horizontal translation of the graph. In , the phase shift is units to the right.
Midline
The horizontal line halfway between the maximum and minimum values. For , the midline is .
Sinusoidal
Having the shape of a sine curve; any function that can be written as a transformed sine or cosine function.

Formula Card

General Form

Complete sine transformation formula

Amplitude

Height from midline to peak

Period

Length of one complete cycle

Phase Shift

Horizontal shift (right if positive)

Vertical Shift

Moves the midline up or down

Key Points

Five critical points for one cycle

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