Define sinusoidal function.
A function resembling sine or cosine, oscillating and periodic.
What is the period (T) of a sinusoidal function?
The length of one complete cycle.
Define frequency (f).
Number of cycles per unit time; reciprocal of period.
What is the midline (k)?
The horizontal line that cuts the wave in half.
Define amplitude.
Vertical distance from midline to max/min value.
What is odd symmetry?
Symmetry about the origin; f(-x) = -f(x).
What is even symmetry?
Symmetry about the y-axis; f(-x) = f(x).
Define oscillation in sinusoidal functions.
Up and down movement between two values, creating a wave.
What does concave up mean?
The curve looks like a smile.
What does concave down mean?
The curve looks like a frown.
Sine vs. Cosine: Symmetry.
Sine: Odd | Cosine: Even
Sine vs. Cosine: Shift.
Cosine is sine shifted left by $\frac{ฯ}{2}$.
Amplitude vs. Midline.
Amplitude: Vertical distance from midline | Midline: Horizontal center.
Period vs. Frequency.
Period: Length of one cycle | Frequency: Cycles per unit time.
Concave Up vs. Concave Down.
Concave Up: Curve smiles | Concave Down: Curve frowns.
What are the differences between period and frequency?
Period: Time for one cycle | Frequency: Cycles per unit time
What are the differences between amplitude and midline?
Amplitude: Vertical distance from midline | Midline: Horizontal center of the wave
What are the differences between sine and cosine functions?
Sine: Starts at midline | Cosine: Starts at maximum/minimum
What are the differences between a vertical stretch and a horizontal stretch?
Vertical: Affects amplitude | Horizontal: Affects period
What are the differences between positive and negative amplitudes?
Positive: Starts increasing | Negative: Starts decreasing
Formula relating cosine and sine.
$cos(ฮธ) = sin(ฮธ + \frac{ฯ}{2})$
Formula for frequency (f).
$f = \frac{1}{T}$
Formula for midline (k).
$k = \frac{y_{max} + y_{min}}{2}$
Formula for amplitude.
$Amplitude = \frac{y_{max} - y_{min}}{2}$
Formula for odd symmetry.
$f(-x) = -f(x)$
Formula for even symmetry.
$f(-x) = f(x)$
Period of standard sine function.
$2\pi$
Period of standard cosine function.
$2\pi$
Frequency of standard sine function.
$\frac{1}{2\pi}$
Frequency of standard cosine function.
$\frac{1}{2\pi}$
