Trigonometric Functions and Graphs
Trigonometric functions sin, cos, and tan can be graphed as continuous waves, and their transformations (amplitude, period, phase shift) model periodic real-world phenomena.
What You Need to Know
Key Concept Diagram
The sine and cosine functions have period 2pi and amplitude 1 in their basic form
Amplitude is the maximum displacement from the midline; it is |a| in y = a sin(bx + c)
Period is the length of one complete cycle; for y = sin(bx) the period is 2pi/b
Phase shift is a horizontal translation; for y = sin(x - d) the shift is d units to the right
Vertical shift moves the midline: y = sin(x) + k shifts the graph up by k units
Key Vocabulary
Amplitude
Half the total vertical range of a trigonometric function; the maximum value above the midline
Period
The horizontal length of one complete cycle of a trigonometric function
Phase shift
A horizontal translation applied to a trigonometric function
Midline
The horizontal axis about which a sinusoidal function oscillates
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the amplitude of y = 3 sin(2x)?
Question 2
What is the period of y = cos(4x)?
Question 3
The graph of y = sin(x - pi/3) is the graph of y = sin(x) shifted by:
Key Concepts Summary
- ●The sine and cosine functions have period 2pi and amplitude 1 in their basic form
- ●Amplitude is the maximum displacement from the midline; it is |a| in y = a sin(bx + c)
- ●Period is the length of one complete cycle; for y = sin(bx) the period is 2pi/b
- ●Phase shift is a horizontal translation; for y = sin(x - d) the shift is d units to the right
- ●Vertical shift moves the midline: y = sin(x) + k shifts the graph up by k units