AP Precalculus · Lesson 3 of 4
Trigonometric and polar functions
Describe amplitude, midline, and period, and interpret polar coordinates as distance and angle.
Sine and cosine graphs
A sine or cosine graph repeats. The midline is the horizontal center of the wave. The amplitude is the distance from the midline to a maximum. The period is the length of one complete repetition, measured in the input’s units, such as radians or seconds.
In y = 3 sin(2x) + 1, the amplitude is 3, the midline is y = 1, and the 2 inside the sine compresses the period from 2π to π. Adding 1 outside changes outputs. Multiplying the input changes how fast the angle increases.
Polar coordinates
A polar point (r, θ) is r units from the origin at an angle θ from the positive x-axis. Negative r is plotted in the opposite direction of θ. The same point can have more than one polar representation because an angle can increase by a full turn.
A polar equation such as r = 4 describes all points 4 units from the origin, which is a circle. Convert with x = r cos θ and y = r sin θ when a Cartesian description is easier.
Worked example
For y = 4 cos(πx) − 2, state the amplitude, midline, and period.
- The amplitude is 4, the coefficient of cosine.
- The midline is y = −2.
- The period is 2π divided by the coefficient of x inside cosine, so 2π / π = 2.
Why this works. Amplitude, midline, and period are three separate features. Do not read the amplitude from the midline.
Check your understanding
- Identify amplitude, midline, and period from an equation and from a graph.
- Explain a horizontal change as a change to the input.
- Plot a polar point using both a distance and an angle.