Advanced Functions

x = a^y is the same as
y = log_a(x)

the logarithmic expression is the inverse of the exponential function

y = alog(k(x-d))+c

Logarithmic Laws

ab^x=log(a)+log(b^x)

log_a(M) = log_a(N) then M = N

log_a(a^x)=x

a ^ log_a(x) = x

log_a(a^-m) = -m

log_a(1/a^m)=-m

log_a(1)=0

log_a(x^n)=nlog_a(x)

log_a(x/y)=log_a(x)-log_a(y)

log_a(xy)=log_a(x)+log_a(y)

3^x=3^3 then x=3

the domain of the log function is {x E r / x > 0}

the range is {y E r}

sound formula in dB is L = 10log(I/I_0)

The compound interest formula is A = P(1 + i)^n

The formula to find pH is pH is pH = -log(H^+)

The formula for the Half-life of an element is N = c(1/2)^t/h

The exponential growth function is y = ab^x

magnitude is the log of the intensity of earthquake to the base of 10

SOHCAHTOA

sec = 1/cos

csc = 1/sin

cot= 1/tan

CAST

only cos is positive in 4th quad, all are positive in 1st quad, only sin is positive in 2nd quad, only tan is positive in 3rd quad

amplitude is the half of the value of the max and min of the function

amplitude is always positive

c value is the equation of axis

equation of axis is the midpoint between the max and min

theta = arc length/radius

angular velocity = ∆θ/∆t

180 degrees= pi

a circle is 2pi

Radians = (pi/180) * angles

degrees = (180/pi) * Radians

Positive a = amplitude

(max-min)/2=a

amplitude is always positive

amplitude is half the distance between max and min

k = 2pi/period

c = equation of axis

(max+min)/2 = eoa

eoa is midpoint of max and min

(positive a )+ c = max
(negative) + c = min

Compound Angle Formulas

sin(a+b)= sin(a)cos(b)+cos(a)sin(b)

sin(a − b) = sin(a) cos (b) − cos(a) sin(b)

cos(a + b) = cos(a) cos(b) − sin(a) sin(b)

cos(a − b) = cos(a) cos(b) + sin(a) sin (b)

tan (a + b) = (tan(a) + tan(b))/(1 - tan(a) * tan(b))

tan (a - b) = (tan(a) - tan(b))/(1 + tan(a) * tan(b))

Double angle formula

sin2θ = 2 sin θ cos θ

cos2θ = cos2θ − sin2θ

tan2θ = 2tanθ / 1 − tan^2 θ

Pythagorean identities

sin^2 θ + cos^2 θ = 1

tan^2 θ + 1 = sec^2 θ

1 + cot^2 θ = csc^2 θ

f(x) =p(x)/q(x) where q(x) ≠ 0

hole occurs at the x value where f(x) = 0/0

the vertical asymptotes: f(x) = p(x)/0

if the degree of p(x) is one higher than the degree of q(x), there is an oblique asymptote

Horizontal asymptote is found by dividing the coefficients of p(x) by the coefficient of q(x)

a/(k(x-d)) + c

solving for x on the bottom results in the vertical asymptote of the equation

if original is linear or quadratic; horizontal asymptote will
be y = 0

the reciprocal will intersect y = 1 or -1 if the original had it in its range

the max and min of original become min and max in reciprocal

the reciprocals of the x intercepts of the original is the x intercept of the reciprocal

the increasing intervals of original turn to decreasing intervals in reciprocals and vice versa

If the polynomial has 2n degrees(n does not = 0), then the two ends of polynomial will open towards same place(top or bottom), if not then they open in different directions

degrees is the highest power value of the polynomial

Do not have vertical of Horizontal asymptotes

Contain only one variable

ordered based on a descending order of powers

When graphed: 0 to n(degrees of polynomial) number of intercepts

the graph has at most n -1 number of turning points

the average rate of change is the rate of change of the graph over an interval

Formula: (f(x_2)-f(x_1))/(x_2-x_1)

Graphically a line must be made between the two points; the slope of the line is the average rate of change

the instantaneous rate of change is the rate of change on an exact point on the graph

(f(x+a)-f(x))/((a+x)-x): a is a small increment from the value of x(usually +0.01)

Tangent line: a line that just touches the exact point and avoids the points on the two sides of the point

Graphically the slope of the line tangent to the point is the instantaneous rate of change

Addition: their y values add up

Subtraction: the y value of the second equation is subtracted from the y value of the first equation

Multiplication: Each term of each Function is multiplied while their degrees are added

Division: use the long division method

When Factoring

First step: Find factors of c for which f(x) = 0

Second step: turn the x = y/z into zx - y = 0

Third step: divide the initial function

Fourth step: Continue process until fully factored

Normally divide the Formula using old school division method

Examples

y=a(k(x-d))^2+c

y = mx+b

y = a2^(k(x-d)) +c

y = asin(k(x-d))+c

y = a(1/(k(x-d))+c

y = a|k(x-d)|+c

y = alog(k(x-d))+c

Transformations

a determines the vertical stretch/compression from the parent function

1/k determines the horizontal stretch or compression from the original function

d determines the horizontal transformation of the function

c determines the vertical transformation of the function

Inverse Functions

switch values of x and y and then solve for x

The domain and range switch as well

Other Important Points

Interval of increase - the intervals in the domain where the y values are getting larger.

Interval of decrease - the intervals in the domain where the y-values are getting smaller.

Odd Function - Function that is the same when rotated 180 degrees around the origin

Even Function - any function that is symmetric about the y-axis

Continuous Function - a function that does not contain any holes or breaks in its graph

Discontinuous Function - a function that contains a hole or break in the graph.