5.1 polynomial functions degree of a polynomial: largest power of x that appears. the zero...
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5.1 Polynomial Functions
Degree of a Polynomial: Largest Power of X that appears. The zero polynomial function f(x) = 0 is not assigned a degree.
Identifying Polynomial Functions
Zeros of Polynomial FunctionsExample: Find a polynomial of degree 3 whose zeros are -4, -2, and 3
For the polynomial, list all zeros and their multiplicities.
Identifying Zeros and Their Multiplicities2 With multiplicity 1-1 with multiplicyty 33 with multiplicity 4
Graphing a Polynomial Using its x-intercepts
(Local Maxima and Local Minima)Leading Coefficient Test for End Behavior
Even-degree polynomial functions have graphs with the same behavior at each end.Odd-degree polynomial functions have graphs with opposite end behavior.Note that the degree is even.
End Behavior : The behavior of a graph for large values of | x |More on End Behavior
A)B)C)D)For the polynomial list the zero and its multiplicity and whether it touches or crosses the x axis at each x-intercept.
Which of the following polynomial functions might have the graph shown?
5.2 & 5.3 Graphing Rational FunctionsRational functions are quotients of polynomial functions.The following are examples of rational functions. Can you find the domains? (All real numbers except values that make the denominator zero)
f(x) = x2 + 7x + 9f(x) = x2 9 g(x) = x h(x) = x + 3 x(x 2)(x + 5) (x 3) (x2 9) x2 + 9Arrow NotationSymbolMeaningx a +x approaches a from the right.x a -x approaches a from the left.x x approaches infinity (x increases without bound)x - x approaches negative infinity (x decreases without bound)
The line x = a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.f (x) as x a + f (x) as x a f (x) as x a + f (x) - as x a -
fayxx = afayxx = af (x) or f(x) - as x approaches a from either the left or the right.Vertical asymptotes occur at all xs where the simplified denominator is zero.fayxx = ax = afayxThe line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound.fyxy = bxyfy = bfyxy = bf (x) b as x f (x) b as x f (x) b as x Finding Horizontal AsymptotesThe degree of the numerator is n. The degree of the denominator is m.If nm,the graph of f has no horizontal asymptote.Note : In contrast to vertical asymptotes, at most 1 horizontal asymptote exists and the graph may cross the horizontal asymptote.
Let: 13Slant/Oblique Asymptotes
-2-1456783217654312-1-3-2Vertical asymptote: x = 3Slant asymptote: y = x - 1A slant asymptote exists if the deegree of the numerator is onemore than the degree of the denominator.
To find the equation of the slant asymptote, divide x - 3 into x2 - 4x 5, and drop the remainder. The asymptote is given by the resulting linear equation.The equation of the slant asymptote is y = x - 1.
Degree 2Degree 131 -4 -5 3 -3
1 -1 -8 Strategy for Graphing a Rational Functionf(x) = p(x) ,where p(x) and q(x) are polynomials with no common factor. q(x)
1. Determine whether the graph of f has symmetry. f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
Find the y-intercept (if there is one) by evaluating f (0). Find the x-intercepts (if there are any) by solving the equation p(x) = 0.
4. Find any vertical asymptote(s) by solving the equation q (x) = 0.5. Find the horizontal asymptote (if there is one)
6. Plot points between and beyond each x-intercept and vertical asymptote.7. Use the information obtained previously to graph the function.
Which of the following is the graph of the equation:A)B)C)D)