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.
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?
\(\omega = \frac{2\pi}{T}\)
Explain how changing 'a' affects the graph.
Changes the amplitude (vertical stretch/compression). Negative 'a' reflects over x-axis.
Explain how changing 'b' affects the graph.
Changes the period (horizontal stretch/compression). Higher 'b' compresses the wave.
Explain how changing 'c' affects the graph.
Shifts the graph horizontally (phase shift). Remember the shift is opposite the sign.
Explain how changing 'd' affects the graph.
Shifts the graph vertically. Also represents the midline of the function.
Why is it important to factor out 'b' before finding the phase shift?
To correctly identify 'c', the horizontal shift, the equation must be in the form $b(\theta + c)$.
Describe the effect of amplitude on the maximum and minimum values of a sinusoidal function.
Amplitude determines how far the maximum and minimum values are from the midline.
How does the period relate to the frequency of a sinusoidal function?
The period is the time for one cycle, while frequency is the number of cycles per unit time. They are inversely proportional.
Explain how the phase shift impacts the starting point of a sinusoidal function.
The phase shift moves the entire wave left or right, changing where the cycle begins relative to the y-axis.
Describe the relationship between the midline and the vertical translation.
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.
How does a reflection over the x-axis affect the shape of the sinusoidal function?
It inverts the function, turning peaks into troughs and vice versa.
