In class practice Sine and Cosine Transformations

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 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 distance along the x-axis required for the function to complete one full cycle. For the function y = sin(kx) or y = cos(kx), the period is given by 2π/k.

Front

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

Back

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

Front

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

Back

The equation is y = 5cos(2x). (Since the period is π, k = 2, and the amplitude is 5.)

Front

What is the phase shift of f(x) = 2sin(3x + π)?

Back

The phase shift is -π/3 units to the left.

Front

What is the vertical shift in the function f(x) = sin(x) + 3?

Back

The vertical shift is 3 units up.

Front

What is the effect of changing the coefficient of x in the function y = sin(kx)?

Back

Changing the coefficient k affects the period of the sine function. The new period is 2π/k.

Front

What is the relationship between sine and cosine functions?

Back

Sine and cosine functions are phase-shifted versions of each other. Specifically, sin(x) = cos(x - π/2) and cos(x) = sin(x + π/2).