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characteristics of polynomial function
 coefficients all real numbers
 degrees are positive integers
 has to have a degree (can be zero/constant function)

domain of a polynomial
all real numbers

degree
largest power of x that appears

name of no degree function
zero function; xaxis

name of 0 degree function
constant function; horizontal line

name of 1 degree function
linear, straight line not horizontal or vertical

name of 2 degree function
quadratic, parabola

parabola opens up/down rule
up if a is positive, down if a is negative

the graph of every polynomial is ?
 smooth: no sharp corners or cusps
 continuous: no gaps or holes

power function
a single monomial taken from a polynomial, like 5x^{2}

important properties of power functions to even integer degrees
 symmetric to yaxis (even)
 domain=all real numbers; range=all nonnegative real numbers

important properties of power functions to odd integer degrees
 symmetric to origin (odd)
 domain and range=all real numbers

theorem you use to find the xintercepts based off factored polynomial
zero product property

xr set to zero represents
xintercepts of the graph; the solutions

multiplicity
exponent to which a factor is raised

sum of the multiplicities =
degree

zeros with even multiplicities ? the xaxis
touch

zeros with odd multiplicities ? the xaxis
cross

turning points
the minima and maxima on a graph, humps and dips

highest number of turning points
degree  1

least degree of a polynomial
turning points + 1

end behavior
behavior of the graph for large values of x, tells you the power function the graph will represent

end behavior of positive leading coefficient and even degree
like a parabola pointing upward, goes to positive infinity on both sides

end behavior of negative leading coefficient and even degree
like a parabola pointing downward, goes to negative infinity on both sides

end behavior of positive leading coefficient and odd degree
like a cube function, towards positive infinity it goes towards positive infinity and towards negative infinity it goes towards negative infinity

end behavior of negative leading coefficient and odd degree
like a cube function flipped over the yaxis; towards positive infinity it goes towards negative infinity and towards negative infinity it goes towards positive infinity

rational numbers
ratios of integers

rational functions
ratios of polynomials, has the form

domain and range of rational functions
 all real numbers except those where the denominator is zero
 to find the domain, set the denominator to zero and solve for the real numbers, these are excluded in the domain

lowest terms
in a rational function, factor out both numerator and denominator and cross out the common factors

how to find intercepts of rational functions
 zeros of the numerator in lowest terms are x intercepts
 solve for r(0) / substitute x for 0 and solve

horizontal asymptote
 the y= line that the values for f(x) approach as the graph goes far to the left and right of the graph (positive/negative infinity)
 sometimes intersects

vertical asymptote
 the x= line that x approaches as the absolute values for r(x) go towards positive infinity
 never intersects

how to find the vertical asymptote of a rational function
in lowest terms, solve for the real zeros of the denominator q(x)

proper
 rational function where the degree of the numerator is less than the degree of the denominator
 e.g. x/x^{2}

improper
rational function where the degree of the numerator is greater than or equal to the degree of the denominator

horizontal asymptote when the rational function is proper (degree of numerator is less than degree of denominator)
y=0

horizontal asymptotes when the rational function has the same degree in numerator as denominator
y=numerator leading coefficient/denominator leading coefficient; a constant function

horizontal asymptotes when the rational function has a numerator with a greater degree than the denominator
use long division to solve for an oblique asymptote, but cannot be anything higher than a linear function

test for symmetry
 solve r(x)
 if the result is the opposite of the original, it has origin symmetry (or, replace x and y with x and y)
 if the result is the same as the original, it has yaxis symmetry (or, replace x with x)

how to find if the graph intersects the horizontal asymptote
Set the function to the constant and solve. The number you get is the intersection as an x intercept coordinate point.

Should the rational function be in lowest terms when you are looking for the domain?
No; exclude holes from the domain as well

How to write the equation of a rational function based off a graph
 Use the x intercepts to get the factors of the numerator, note also if the graph touches (even) or crosses (odd)
 Look at how the graph looks around the vertical asymptotes, determine these and put in the denominator as (x#)
 If it looks like a reciprocal function (the up/down ends of the graph don't go in the same direction), the factor is odd.
 If the two ends go in the same direction (1/x^{2}), then the factor is even.
 use the horizontal asymptote to determine the correct ratio of degrees and any coefficients


(a^{3}+b^{3})
(a+b)(a^{2}ab+b^{2})

(a^{3}b^{3})
(ab)(a^{2}+ab+b^{2})

(a+b)^{2}
a^{2}+2ab+b^{2}

(ab)^{2}
a^{2}2ab+b^{2}

how to solve a polynomial inequality using its graph
 solutions include the numbers that make the function positive
 look for where the graph is above the xaxis
 use intercepts to determine where the number line is cut up, then show the intervals between these for where the function is positive in interval notation

how to solve a polynomial inequality algebraically
 rewrite so that the polynomial expression is on the left and zero is on the right
 find the real zeros by factoring, use these to divide the number line into intervals
 chose a number in each interval and do a test by plugging it into the function, and see if it comes out as a positive or negative value
 whether or not it is included in the solution set depends on the sign of the inequality

how to solve a rational inequality using its graph
 use the vertical asymptotes and the xintercepts to divide the number line into intervals
 intervals that are a solution to the function show the graph above the xaxis (positive y values)

how to solve a rational inequality algebraically
 rewrite so that f is on the left and zero is on the right
 find the x intercepts and vertical asymptotes by finding the solutions to the numerator (intercepts) and denominator (asymptotes)
 use these to divide the number line into intervals, chose a point within each interval and plug it in for x, then take note of whether the solution is positive or negative
 refer to the sign of the inequality
 in the solution set, do not include values of the vertical asymptotes or holes

should you check for symmetry on the original equation or lowest terms?
original

how to identify a hole
 in lowest terms, sometimes a factor crosses out
 set this to zero and this gives you the x coordinate
 plug x coordinate into the lowest terms expression and solve for y coordinate

y=f(x)+k
raise by k units

y=f(x)k
lower by k units

y=f(x+h)
left shift h units

y=f(xh)
right shift h units

vertical compression/stretch function transformation
 f(x) > af(x); multiply the function by a
 a<1 means compression
 a>1 means stretch

changes to the coordinate pairs for a vertical compression/stretch
multiply the ycoordinate by a

graphically, what does a vertical compression/stretch look like
 stretch: pulling it up
 compression: pushing it down

horizontal compression/stretch function transformation
 f(x) > f(ax); replace x with ax
 a<1 means stretch
 a>1 means compression

changes to the coordinate pairs for horizontal compression/stretch
multiply the xcoordinate by 1/a

graphically, what does a horizontal compression/stretch look like
 stretch: pulling the sides out
 compression: pushing the sides together

xaxis reflection transformation
 flipped up or below horizontal axis
 f(x) > f(x); multiply the function by 1

yaxis reflection transformation
 flipped to left or right over the vertical axis
 f(x) > f(x); replace x with x in the function

