$T = \frac{2\pi}{b}$

$f(\theta) = a\sin(b(\theta + c)) + d$

$f(\theta) = a\cos(b(\theta + c)) + d$

$d = \frac{max + min}{2}$

$a = \frac{max - min}{2}$

$b = \frac{2\pi}{T}$

Ensure the equation is in the form $b(\theta + c)$, then the phase shift is -c.

$f = \frac{1}{T}$

$\omega = b$

$\omega = \frac{2\pi}{T}$

1. Find midline (d). 2. Find amplitude (a). 3. Find period (T), then calculate b. 4. Determine phase shift (c). 5. Write the equation.

1. Identify the midline (d). 2. Identify the amplitude (a). 3. Max = d + a, Min = d - a.

1. Check if the points are consecutive peaks or troughs. 2. If so, the horizontal distance between them is the period. 3. If not, analyze the points to determine what fraction of the period they represent.

1. Identify a key point (max, min, or midline crossing). 2. Determine how far this point is shifted horizontally from the standard sine or cosine function. 3. Use this shift to find 'c'.

1. Find the midline (d) and amplitude (a) from max and min. 2. Find 'b' from the period. 3. Use the given point to solve for the phase shift 'c'.

1. Recall that $T = \frac{1}{f}$. 2. Substitute the given frequency into the formula. 3. Calculate the value of T.

1. If the function starts at the midline, use sine. 2. If it starts at a maximum or minimum, use cosine. 3. Adjust the phase shift as needed.

1. Recall that $T = \frac{2\pi}{b}$. 2. Rearrange to solve for b: $b = \frac{2\pi}{T}$. 3. Substitute the period and calculate.

1. Find the other parameters as usual. 2. Include a negative sign in front of the 'a' value.

1. Check the amplitude, period, phase shift, and vertical shift. 2. Plug in key x-values (e.g., x=0, x=T/4, x=T/2) to see if the y-values match the graph.

Changes the amplitude (vertical stretch/compression). Negative 'a' reflects over x-axis.

Changes the period (horizontal stretch/compression). Higher 'b' compresses the wave.

Shifts the graph horizontally (phase shift). Remember the shift is opposite the sign.

Shifts the graph vertically. Also represents the midline of the function.

To correctly identify 'c', the horizontal shift, the equation must be in the form $b(\theta + c)$.

Amplitude determines how far the maximum and minimum values are from the midline.

The period is the time for one cycle, while frequency is the number of cycles per unit time. They are inversely proportional.

The phase shift moves the entire wave left or right, changing where the cycle begins relative to the y-axis.

The midline is the horizontal line that runs midway between the maximum and minimum values, and the vertical translation shifts the midline up or down.

It inverts the function, turning peaks into troughs and vice versa.