Polynomial & Rational Functions

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

all real numbers

largest power of x that appears

zero function; x-axis

constant function; horizontal line

linear, straight line not horizontal or vertical

quadratic, parabola

up if a is positive, down if a is negative

• smooth: no sharp corners or cusps
• continuous: no gaps or holes

a single monomial taken from a polynomial, like -5x²

• symmetric to y-axis (even)
• domain=all real numbers; range=all nonnegative real numbers

• symmetric to origin (odd)
• domain and range=all real numbers

zero product property

x-intercepts of the graph; the solutions

exponent to which a factor is raised

degree

touch

cross

the minima and maxima on a graph, humps and dips

degree - 1

turning points + 1

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

like a parabola pointing upward, goes to positive infinity on both sides

like a parabola pointing downward, goes to negative infinity on both sides

like a cube function, towards positive infinity it goes towards positive infinity and towards negative infinity it goes towards negative infinity

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

ratios of integers

ratios of polynomials, has the form

• 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

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

• zeros of the numerator in lowest terms are x intercepts 
• solve for r(0) / substitute x for 0 and solve

• 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

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

in lowest terms, solve for the real zeros of the denominator q(x)

• rational function where the degree of the numerator is less than the degree of the denominator 
• e.g. x/x²

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

y=0

y=numerator leading coefficient/denominator leading coefficient; a constant function

use long division to solve for an oblique asymptote, but cannot be anything higher than a linear function

• 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)

Set the function to the constant and solve. The number you get is the intersection as an x intercept coordinate point.

No; exclude holes from the domain as well

• 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

(a+b)(a²-ab+b²)

(a-b)(a²+ab+b²)

a²+2ab+b²

a²-2ab+b²

• 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

• 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

• 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)

• 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

original

• 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

raise by k units

lower by k units

left shift h units

right shift h units

• f(x) --> af(x); multiply the function by a
• |a|<1 means compression
• |a|>1 means stretch

multiply the y-coordinate by a

• stretch: pulling it up
• compression: pushing it down

• f(x) --> f(ax); replace x with ax
• |a|<1 means stretch
• |a|>1 means compression

multiply the x-coordinate by 1/a

• stretch: pulling the sides out
• compression: pushing the sides together

• flipped up or below horizontal axis
• f(x) --> -f(x); multiply the function by -1

• flipped to left or right over the vertical axis
• f(x) --> f(-x); replace x with -x in the function