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Gymnastxoxo17
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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)
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domain of a polynomial
all real numbers
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degree
largest power of x that appears
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name of no degree function
zero function; x-axis
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name of 0 degree function
constant function; horizontal line
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name of 1 degree function
linear, straight line not horizontal or vertical
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name of 2 degree function
quadratic, parabola
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parabola opens up/down rule
up if a is positive, down if a is negative
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the graph of every polynomial is ?
- smooth: no sharp corners or cusps
- continuous: no gaps or holes
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power function
a single monomial taken from a polynomial, like -5x2
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important properties of power functions to even integer degrees
- symmetric to y-axis (even)
- domain=all real numbers; range=all nonnegative real numbers
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important properties of power functions to odd integer degrees
- symmetric to origin (odd)
- domain and range=all real numbers
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theorem you use to find the x-intercepts based off factored polynomial
zero product property
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x-r set to zero represents
x-intercepts of the graph; the solutions
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multiplicity
exponent to which a factor is raised
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sum of the multiplicities =
degree
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zeros with even multiplicities ? the x-axis
touch
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zeros with odd multiplicities ? the x-axis
cross
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turning points
the minima and maxima on a graph, humps and dips
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highest number of turning points
degree - 1
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least degree of a polynomial
turning points + 1
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end behavior
behavior of the graph for large values of x, tells you the power function the graph will represent
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end behavior of positive leading coefficient and even degree
like a parabola pointing upward, goes to positive infinity on both sides
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end behavior of negative leading coefficient and even degree
like a parabola pointing downward, goes to negative infinity on both sides
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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
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end behavior of negative leading coefficient and odd degree
like a cube function flipped over the y-axis; towards positive infinity it goes towards negative infinity and towards negative infinity it goes towards positive infinity
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rational numbers
ratios of integers
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rational functions
ratios of polynomials, has the form
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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
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lowest terms
in a rational function, factor out both numerator and denominator and cross out the common factors
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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
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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
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vertical asymptote
- the x= line that x approaches as the absolute values for r(x) go towards positive infinity
- never intersects
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how to find the vertical asymptote of a rational function
in lowest terms, solve for the real zeros of the denominator q(x)
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proper
- rational function where the degree of the numerator is less than the degree of the denominator
- e.g. x/x2
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improper
rational function where the degree of the numerator is greater than or equal to the degree of the denominator
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horizontal asymptote when the rational function is proper (degree of numerator is less than degree of denominator)
y=0
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horizontal asymptotes when the rational function has the same degree in numerator as denominator
y=numerator leading coefficient/denominator leading coefficient; a constant function
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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
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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 y-axis symmetry (or, replace x with -x)
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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.
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Should the rational function be in lowest terms when you are looking for the domain?
No; exclude holes from the domain as well
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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/x2), then the factor is even.
- use the horizontal asymptote to determine the correct ratio of degrees and any coefficients
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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 x-axis
- 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
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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
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how to solve a rational inequality using its graph
- use the vertical asymptotes and the x-intercepts to divide the number line into intervals
- intervals that are a solution to the function show the graph above the x-axis (positive y values)
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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
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should you check for symmetry on the original equation or lowest terms?
original
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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
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y=f(x)+k
raise by k units
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y=f(x)-k
lower by k units
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y=f(x+h)
left shift h units
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y=f(x-h)
right shift h units
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vertical compression/stretch function transformation
- f(x) --> af(x); multiply the function by a
- |a|<1 means compression
- |a|>1 means stretch
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changes to the coordinate pairs for a vertical compression/stretch
multiply the y-coordinate by a
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graphically, what does a vertical compression/stretch look like
- stretch: pulling it up
- compression: pushing it down
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horizontal compression/stretch function transformation
- f(x) --> f(ax); replace x with ax
- |a|<1 means stretch
- |a|>1 means compression
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changes to the coordinate pairs for horizontal compression/stretch
multiply the x-coordinate by 1/a
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graphically, what does a horizontal compression/stretch look like
- stretch: pulling the sides out
- compression: pushing the sides together
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x-axis reflection transformation
- flipped up or below horizontal axis
- f(x) --> -f(x); multiply the function by -1
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y-axis reflection transformation
- flipped to left or right over the vertical axis
- f(x) --> f(-x); replace x with -x in the function
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