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10 Cards in this Set
- Front
- Back
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Rational function
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A function of the form
r(x) = P(x)/Q(x) where P and Q are polynomials |
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Definition of a vertical asymptote
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The line x = a is a vertical asymptote of the function y = f(x) if y approaches +/- ∞ as x approaches a from the left or right
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Definition of a horizontal asymptote
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The line y = b is a horizontal asymptote of the function y = f(x) if y approaches b as x approaches +/- ∞
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Sketching Graphs of Rational Functions
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1) Factor the numerator and denominator.
2) Find the x-intercepts by determining the zeros of the numerator, and the y-intercept from the value of the function at x = 0. 3) Find the vertical asymptotes by determining the zeros of the denominator. 4) Find the horizontal asymptote (if any) by dividing both numerator and denominator by the highest power of x that appears in the denominator. - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is 0 - If the degree of the numerator is more than the degree of the denominator, there is no horizontal asymptote - If the degree of both the numerator and the denominator are the same, the horizontal asymptote is the quotient of the numerator's highest degree's coefficient divided by the denominator's highest degree's coefficient |
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Continuous function
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A function with a graph that consists of a single unbroken curve
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What are the first two steps for graphing polynomial functions?
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1) Factor the polynomial to find all its real zeros; these are the x-intercepts of the graph.
2) Find the y-intercept (x = 0). |
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If c is a zero of P with an odd multiplicity, then what is the shape of its graph?
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The graph crosses the x-axis.
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If c is a zero of P with an even multiplicity, then what is the shape of its graph?
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The graph does not cross the x-axis.
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In A(x - c)^m, what is the multiplicity?
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m
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Cut points
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If r(x) = P(x)/Q(x) is a rational function, then the cut points for r are the values of x at which either P(x) or Q(x) is 0.
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