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Math1050 Ch05

Polynomial and Rational Functions

TermDefinition
T or F? Polynomial functions must have powers of nonnegative integers. T
Graphs of polynomial functions must be both ___ and ___. continuous, smooth
If f(r)=0 then r is a ___. real zero
Real zeros are ___-intercepts. x
For real zeros r, ___ is a factor of f. x-r
For real zeros r, r is a solution to f(x)=___. 0
Power functions are a type of ___ function. polynomial
Power functions are in the form f(x)=___. ax^n
f(x)=3x is a power function of degree ___. It is a ___. Its y-intercept is ___. 1, straight line, 0
f(x)=-5x^2 is a power function of degree ___. It is a ___. Its y-intercept is ___. Vertex is at ___. It opens ___. 2, parabola, 0, (0,0), down
For power functions of even degree, the domain is ___ and the range is ___. all real numbers, all nonnegative numbers
For power functions of even degree, the graph always contains points ___, ___, ___. origin, (-1,1), (1,1)
For power functions of odd degree, the graph is symmetric to the ___. origin
For power functions of even degree, the graph is symmetric to the ___. y-axis
For power functions of odd degree, the domain is ___ and the range is ___. all real numbers, all real numbers
For power functions of odd degree, the graph always contains points ___, ___, ___. origin, (-1,-1), (1,1)
For zeros of even multiplicity, the graph ___ the x-axis. touches
For zeros of odd multiplicity, the graph ___ the x-axis. crosses (hint: cross-dressers are odd)
If f is a polynomial function of degree 3, the graph has AT MOST ___ turning points. 2, (could be fewer though), i.e., n-1 TPs
For large values of |x|, f(x)=ax^5+bx^4+... resembles the graph ___. This is called its ___. y=ax^5, end behavior
For large values of |x|, f(x)=x^2(x-2) resembles the graph ___. y=x^3
Analyzing the graph of a polynomial function: Step 1 Determine the end behavior, rewrite in standard form if necessary
Analyzing the graph of a polynomial function: Step 2 Find the x & y intercepts
Analyzing the graph of a polynomial function: Step 3 determine the zeros and their multiplicity and determine whether they cross or touch the x-axis
Analyzing the graph of a polynomial function: Step 4 determine the max TPs
Analyzing the graph of a polynomial function: Step 5 determine the behavior near each zero
Analyzing the graph of a polynomial function: Step 6 draw
Ratios of polynomials are called ___. rational functions
Rational function are said to be in lowest terms if they have no ___. common factors
T or F? Asymptotes are always linear functions. T
___ asymptotes are the reals zeros of the denominator. Vertical
To find asymptotes, the function must first be in ___. lowest terms
In lowest terms, a rational fxn is said to be ___ if the degree of the numerator is less than that of the denominator. proper
All proper functions have the horizontal asymptote ___. y=0
In lowest terms, a rational fxn is said to be ___ if the degree of the numerator equal to or more than that of the denominator. improper
To find horizontal and oblique asymptotes of improper fxns, we use long division to write the rational fxn as the ___ of the polynomial (the quotient) plus a proper rational fxn (the remainder over the denominator). sum,
If the polynomial quotient of a rational fxn is f(x)=b, a constant, then the line ___ is a ___ asymptote. y=b, horizontal
If the polynomial quotient of a rational fxn is f(x)=ax+b, then the line ___ is a ___ asymptote. y=ax+b, oblique
In lowest terms, if the numerator and denominator are of the same degree in a rational fxn, the quotient of the ___ will give us the ___ asymptote. leading coefficients, horizontal
In lowest terms, if the quotient of a rational function is not ___, it has neither a horizontal nor oblique asymptote. linear
Steps for finding horizontal & oblique asymptotes: The rational fxn must first be ___. in lowest terms
Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n<m, R is ___ and it has a/an ___ asymptote which is ___. proper, horizontal, y=0
Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n=m, R is ___ and it has a/an ___ asymptote which is ___. improper, horizontal, y=a/b (the ratio of the leading coefficients)
Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n=m+1, R is ___ and it has a/an ___ asymptote which is ___. improper, oblique, in the form y=ax+b
Steps for finding horizontal & oblique asymptotes (For R(x), n=degree of numerator, m=degree of denominator): If n = m + 2 or more, R is ___ and it has ___ asymptote and the graph behaves like the graph of the ___ for large values of |x|. improper, no horizontal or oblique, quotient
The graph of a rational fxn has either ___ horizontal or ___ oblique asymptote or it has ___. 1, 1, neither
Analyzing the graph of a rational fxn R: Step 1 Factor the fxn and find the domain of the denominator.
Analyzing the graph of a rational fxn R: Step 2 Write R in lowest terms
Analyzing the graph of a rational fxn R: Step 3 Locate the intercepts of R, determine the end behavior at each x-intercept
The x-intercepts of rational fxn R are the zeros of the ___ that are in the domain of R. numerator
Analyzing the graph of a rational fxn R: Step 4 Determine and graph the vertical asymptotes
Analyzing the graph of a rational fxn R: Step 5 Determine and graph the horizontal or oblique asymptote (if there is one). Determine if the graph intersects the asymptote. Plot these points.
Analyzing the graph of a rational fxn R: Step 6 Use the zeros to divide the x-axis into intervals. Determine where the graph is above and below by plotting points.
Analyzing the graph of a rational fxn R: Step 7 Analyze and graph the behavior near each asymptote.
Analyzing the graph of a rational fxn R: Step 8 Graph
Solving a polynomial inequality: Step 1 Write the inequality with a zero on one side. i.e., f(x)<0, f(x)>0, f(x)< or = 0, f(x)> or = 0
Solving a polynomial inequality: Step 2 Determine the zeros and if it is rational, where the fxn is undefined.
Solving a polynomial inequality: Step 3 Separate x into intervals.
Solving a polynomial inequality: Step 4 Determine where f(x) is + or - for each interval. BE CAREFUL TO EXCLUDE WHERE f IS UNDEFINED.
If f(r)=0 then ___ is an x-intercept, ___ is a factor of f, and r is a solution to the equation ___. r, x-r, f(x)=0
The remainder theorem says the if f(x) is divided by x-c, then the remainder is ___. f(c)
The factor theorem says that: 1. If f(c)=0 then ___ is a factor of ___. 2. If ___ is a factor of f(x), then ___. x-c, f(x), x-c, f(c)=0
A polynomial function cannot have more real zeros than ___. its degree
Rational zeros theorem: For a polynomial fxn R in standard form, if p/q is a factor of R then p must be a factor of ___ and q must be a factor of ___. The last coefficient, the leading coefficient
T or F? At least one of the possible zeros from the list of the possible rational zeros must be a zero. T
The quotient of a fxn R and a factor is called a ___ equation. depressed
f(x)= x^3 + 3x^2 + 4x + 12 can have at most ___ zeros. 3
For f(x)= 2x^3 + 3x^2 + 4x + 12, possible zeros are ___. + or - (1, 2, 3, 4, 6, 12, 1/2, 3/2)
Intermediate value theorem says that if a<b and f(a) and f(b) are of ___ sign, there is at least one real zero between a and b. opposite
T or F? Complex polynomials have the same number of linear factors as the degree. T
For complex polynomials, if r=a+bi is a zero then ___ is also a zero. This is called its ___. a-bi, complex conjugate
(a-bi)(a+bi)=___ a^2 + b^2
A polynomial of ___ degree with real coefficients has at least one real zero. odd
Created by: drjolley