4.02 The Basics of Sine and Cosine Graphs

Front

What is the period of a sine or cosine function?

Back

The period is the distance (in the x-direction) it takes for the function to complete one full cycle. For sine and cosine functions, the period is given by the formula: Period = 2π / |b|, where b is the coefficient of x in the function y = a sin(bx) or y = a cos(bx).

Front

What is the midline of a sine or cosine graph?

Back

The midline is a horizontal line that represents the average value of the maximum and minimum values of the function. It can be found by taking the average of the maximum and minimum values of the function.

Front

How do you determine the range of a sine or cosine function?

Back

The range of a sine or cosine function is determined by the amplitude and the midline. If the midline is y = d and the amplitude is a, the range is [d - a, d + a].

Front

What is the amplitude of a sine or cosine function?

Back

The amplitude is the maximum distance from the midline to the peak (or trough) of the graph. It is given by the absolute value of the coefficient a in the function y = a sin(bx) or y = a cos(bx).

Front

If the range of a sine wave is [-10, -2], what is the amplitude?

Back

The amplitude is 4, calculated as (max - min) / 2 = (-2 - (-10)) / 2 = 4.

Front

What is the formula for the sine function?

Back

The sine function is defined as y = a sin(bx + c) + d, where a is the amplitude, b affects the period, c is the phase shift, and d is the vertical shift.

Front

What is the formula for the cosine function?

Back

The cosine function is defined as y = a cos(bx + c) + d, where a is the amplitude, b affects the period, c is the phase shift, and d is the vertical shift.

Front

How do you find the key points of a sine or cosine wave?

Back

Key points can be found by determining the x-values at which the function reaches its maximum, minimum, and midline. For a sine wave, these points occur at intervals of π/2.

Front

What is half of π/8?

Back

π/16.

Front

How many key points are required when sketching a sine or cosine wave?

Back

5 key points are typically required to accurately sketch a sine or cosine wave.