Advanced Functions

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Advanced Functions

Characteristics of Functions

a function is a graph that does not have two y values for one x value

Other Important Points

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

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

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

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

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

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

Inverse Functions

The domain and range switch as well

switch values of x and y and then solve for x

Transformations

c determines the vertical transformation of the function

d determines the horizontal transformation of the function

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

a determines the vertical stretch/compression from the parent function

Examples

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

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

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

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

y = mx+b

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

Operations on Functions

Division: use the long division method

Normally divide the Formula using old school division method

When Factoring

Fourth step: Continue process until fully factored

Third step: divide the initial function

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

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

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

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

Addition: their y values add up

Average and Instantaneous 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)

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

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

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

Polynomial and Rational Function

Polynomials

A function of the form f(x) = a_n (x)^n + a_(n-1) (x)^(n-1). . . .

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

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

ordered based on a descending order of powers

Contain only one variable

Do not have vertical of Horizontal asymptotes

degrees is the highest power value of the polynomial

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

Reciprocal Function

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

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

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

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

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

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

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

Rational Functions

f(x) =p(x)/q(x) where q(x) ≠ 0

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

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

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

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

Trigonometry

Trigonometric equations

Major identities

Pythagorean identities

1 + cot^2 θ = csc^2 θ

tan^2 θ + 1 = sec^2 θ

sin^2 θ + cos^2 θ = 1

Double angle formula

tan2θ = 2tanθ / 1 − tan^2 θ

cos2θ = cos2θ − sin2θ

sin2θ = 2 sin θ cos θ

Compound Angle Formulas

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

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

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

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

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

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

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

Tangent Graph

cos(x)

sin(x)

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

c = equation of axis

(max+min)/2 = eoa

eoa is midpoint of max and min

k = 2pi/period

Positive a = amplitude

amplitude is half the distance between max and min

(max-min)/2=a

Radians

degrees = (180/pi) * Radians

Radians = (pi/180) * angles

a circle is 2pi

180 degrees= pi

angular velocity = ∆θ/∆t

theta = arc length/radius

the Connection between sin, cos, x, y, degrees, radians, and the graph

Recall from Functions 11

equation of axis is the midpoint between the max and min

c value is the equation of axis

amplitude is always positive

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

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

cot= 1/tan

csc = 1/sin

sec = 1/cos

SOHCAHTOA

Logarithms and Exponents

Applicable Logarithmic Formulas

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

The exponential growth function is y = ab^x

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

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

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

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

What is a Logarithm

the parent log function

the range is {y E r}

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

log function, but decreasing

Logarithmic Laws

3^x=3^3 then x=3

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

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

log_a(x^n)=nlog_a(x)

log_a(1)=0

log_a(1/a^m)=-m

log_a(a^-m) = -m

a ^ log_a(x) = x

log_a(a^x)=x

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

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

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

the logarithmic expression is the inverse of the exponential function

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