How can you determine whether a graph represents a parent sine function or a parent cosine function?

In relation to the y-axis the sine wave is in blue and the cosine is in red.

The sine function is called odd because it has rotational symmetry of 180°.

The cosine function is called even because it is reflected about the y-axis.

sin Θ = -sin -Θ; cos Θ = cos -Θ

sin Θ = cos (Θ - π/2) = cos (Θ - 90°)

cos Θ = sin (Θ + π/2) = sin (Θ + 90°)

A negative sign in front of the function inverts it and a number magnifies its y-value.

A number added or subtracted to the function moves it up or down respectively.

A coefficient of theta greater than 1 condenses the waveform and between 1 and zero stretches it out in the x-direction. If the coefficient is negative, it flips it across the y-axis.

A negative constant inside the function shifts the function to the right and a positive constant inside the function shifts it to the left.

In relation to the y-axis the sine wave is in blue and the cosine is in red.

Below is a transformation:

Red is y = sin Θ

Orange is y = sin (Θ + π/4) Notice the red is shifted to the left.

Green is y = sin (2Θ + π/4) Notice the frequency of the orange is doubled and wavelength is halved.

Blue is y = sin (2Θ + π/4) +2 Notice the green is shifted up by 2.

Black is y = ½ sin (2Θ + π/4) + 2 The amplitude of blue is halved.