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Rational Functions and Their Graphs
Section 3.6 Part 2 Rational Functions and Their Graphs Objectives: Graph rational functions. Identify slant asymptotes. Solve applied problems involving rational functions.
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Strategy for Graphing a Rational Function
Suppose that f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions with no common factors. 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. Find any vertical asymptote(s) by solving the equation q (x) = 0. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. Plot at least one point between and beyond each x-intercept and vertical asymptote. Use the information obtained previously to graph the function between and beyond the vertical asymptotes.
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Sketch the graph of: Symmetry y-intercept x –intercept Vertical asymptote Horizontal asymptote Test points
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the graph has no symmetry The y-intercept is (0,-3/10)
Solution: the graph has no symmetry The y-intercept is (0,-3/10) The x-intercept is (3/2, 0) The vertical asymptote is x = -2 The horizontal asymptote is y = 2/5 Test points include (-3, 9/5), (0, -3/10), (2, 1/20)
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Practice #1 Graph:
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Practice #1 (cont.)
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Practice #2 Graph:
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Practice #2 (cont.)
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Text Example Find the slant asymptote of
Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x - 3 into x2 - 4x If it cannot be factored, use long division. Ignore any remainders when determining the slant asymptote. The equation of the slant asymptote is y = x - 1. Using our strategy for graphing rational functions, the graph is shown. -2 -1 4 5 6 7 8 3 2 1 -3 Vertical asymptote: x = 3 Slant asymptote: y = x - 1
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Practice #4 Find the slant asymptote of
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Real Life Application The average cost per unit for a company to produce x units is the sum of its fixed and variable costs divided by the number of units produced. The average cost function is modeled below: A company is planning to manufacture wheelchairs that are light, Fast, and beautiful. Fixed monthly cost sill be $500,00, and it will Cost $400 to produce each radically innovative chair. (a) Write the average cost function of producing x wheelchairs.
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(b) Find and interpret:
(c) What is the horizontal asymptote for the average cost function? Describe what this represents for the company. (d) Why would this model create difficulties for small businesses?
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