What is the formula for the Period (T) of a sinusoidal function?
$T = \frac{2\pi}{b}$
General form of a sine function?
$f(\theta) = a\sin(b(\theta + c)) + d$
General form of a cosine function?
$f(\theta) = a\cos(b(\theta + c)) + d$
How to calculate the midline (d) given the max and min values?
$d = \frac{max + min}{2}$
How to calculate the amplitude (a) given the max and min values?
$a = \frac{max - min}{2}$
Formula to find 'b' given the period T?
$b = \frac{2\pi}{T}$
How to find the phase shift from the equation?
Ensure the equation is in the form $b(\theta + c)$, then the phase shift is -c.
What is the formula for frequency (f) given the period (T)?
$f = \frac{1}{T}$
What is the relationship between \(\omega\) (angular frequency) and b?
\(\omega = b\)
What is the formula for \(\omega\) (angular frequency) given period T?
\(\omega = \frac{2\pi}{T}\)
What is the definition of Amplitude?
The height of the wave from its midline, always a positive value.
What is the definition of Period?
The length of one complete cycle of the wave.
What is the definition of Phase Shift?
The horizontal shift of the wave (left or right).
What is the definition of Vertical Translation?
The vertical shift of the wave (up or down); also the midline.
Define Sinusoidal Function.
Wave-like graphs based on sine and cosine, exhibiting periodic behavior.
What does a negative 'a' value indicate in a sinusoidal function?
A reflection of the wave over the x-axis.
What does the parameter 'b' affect in a sinusoidal function?
The period (horizontal stretch or compression) of the wave.
How does a positive 'c' value affect the phase shift?
Shifts the wave to the left.
What does the 'd' value represent in a sinusoidal function?
The vertical translation or midline of the wave.
What is the relationship between period and frequency?
Period is the inverse of frequency.
How to find the equation from a graph (sinusoidal)?
1. Find midline (d). 2. Find amplitude (a). 3. Find period (T), then calculate b. 4. Determine phase shift (c). 5. Write the equation.
How to find the maximum and minimum values given the equation?
1. Identify the midline (d). 2. Identify the amplitude (a). 3. Max = d + a, Min = d - a.
Given two points on a sinusoidal graph, how do you determine the period?
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.
How to determine the phase shift if the starting point is not on the y-axis?
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'.
How do you determine the equation of a sinusoidal function given its maximum value, minimum value, and a point it passes through?
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'.
How to find the period if given frequency?
1. Recall that \(T = \frac{1}{f}\). 2. Substitute the given frequency into the formula. 3. Calculate the value of T.
How to determine if a sinusoidal function should be modeled with sine or cosine?
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.
How to find the 'b' value from the period?
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.
How do you find the equation of a sinusoidal function if it is reflected over the x-axis?
1. Find the other parameters as usual. 2. Include a negative sign in front of the 'a' value.
How do you verify that a sinusoidal equation matches a given graph?
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.
