Sine and Cosine Graphs

Front

What is the sine function?

Back

The sine function, denoted as sin(x), is a periodic function that describes the y-coordinate of a point on the unit circle as it rotates around the origin. It has a range of [-1, 1] and a period of 2π.

Front

What is the cosine function?

Back

The cosine function, denoted as cos(x), is a periodic function that describes the x-coordinate of a point on the unit circle as it rotates around the origin. It has a range of [-1, 1] and a period of 2π.

Front

What is the amplitude of a sine or cosine function?

Back

The amplitude is the maximum distance from the midline (equilibrium position) of the graph to its peak or trough. It is determined by the coefficient in front of the sine or cosine function.

Front

What is the period of a sine or cosine function?

Back

The period is the length of one complete cycle of the wave. For the functions y = sin(kx) or y = cos(kx), the period is given by 2π/k.

Front

How do you find the phase shift of a sine or cosine function?

Back

The phase shift is determined by the horizontal translation of the function. For y = sin(k(x - d)) or y = cos(k(x - d)), the phase shift is d units to the right if d is positive and to the left if d is negative.

Front

What is the vertical shift in a sine or cosine function?

Back

The vertical shift is the amount by which the graph is moved up or down. It is determined by the constant added or subtracted from the function, e.g., y = sin(x) + c shifts the graph c units up.

Front

What is the equation for a sine function with an amplitude of 3 and a period of π?

Back

y = 3sin(2x) (since the period is 2π/k, k must be 2 for the period to be π).

Front

What is the equation for a cosine function with an amplitude of 5 and a period of 2π/3?

Back

y = 5cos(3x) (since the period is 2π/k, k must be 3 for the period to be 2π/3).

Front

What is the effect of a negative sign in front of a cosine function?

Back

A negative sign in front of a cosine function, e.g., y = -cos(x), reflects the graph across the x-axis.

Front

What is the effect of changing the coefficient of x in a sine function?

Back

Changing the coefficient of x in a sine function, e.g., y = sin(kx), affects the period of the function. The period is given by 2π/k.