In what way does the phase shift affect the graph?
The phase shift is the horizontal shift.
In the form y = A sin(B x + C) + D the phase shift is - C / B.
In the form y = A sin(B ( x - phi) ) + D the phase shift is phi.
Is the amplitude solely determined by the coeffeciant of the sin, cos, or tan?
The tangent function does not have an amplitude, since it has vertical asymptotes.
The sine and cosine functions do have amplitude. The amplitude is the coefficient of the sine or cosine, and it is solely determined by that coefficient.
This question refers to Precalculus I, but I did not quite grasp the laws of exponents. What are the refered to and how are they used?
The following is a logically sufficient statement of the laws of exponents:
(xy)^n = x^n * y^n.
x^n * x^m = x^(n + m).
x^(-n) = 1 / x^n.
(x^n)^m = x^(n * m).
x^0 = 1.
Also included in some statements of the laws, even though they follow immediately from the above set:
(x / y) ^ n = x^n / y^n.
x^n / x^m = x^(n - m).
We use these laws throughout algebra.
Is there a trick to memorizing the unit circle that will make estimations of arc lengths and angles more precise?
If you go around a circle in 30 degree increments you have 12 such increments, 3 for each quadrant.
30 degrees is pi/6 radians. So just label the circle in increments of 30 degrees, then replace by pi/6, 2 pi/6, 3 pi/6, etc. Finally reduce any fractions in your expressions (e.g., 3 pi/6 = pi/2; 4 pi/6 = 2 pi / 3; etc.).
45 degrees is pi / 4 radians. So just label the circle in increments of 45 deg, then replace by pi/4, 2 pi/4, 3 pi/4 etc. and when possible reduce fractions.