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Unformatted text preview: Graphing Rational Functions and Solving Inequalities Math 150
Rational functions behave differently than polynomial functions. However, there are a few things which they have in common. Let R⎛ x ⎞ = ⎝⎠
p ⎛ x⎞ ⎝ ⎠ . Where will the xintercepts occur? Where will the yintercept occur? q⎝ x⎞ ⎛⎠ Let's look at the basic function f ⎛ x ⎞ = ⎝⎠ 1 . x
10 8 6 4 2 10 8 6 4 2 2 4 6 8 10 2 4 6 8 10 What do you notice about the graph of this function? Where is
1 > 0 ? ⎛ 0, ∞⎞ ⎝ ⎠ ⎜ ⎟ x Where is 1 < 0 ? ⎛  ∞, 0⎞ ⎝ ⎠ ⎜ ⎟ x Now let's get a little more complicated. Without using the calculator, draw the graph of ⎛ x + 1⎞ ⎝ ⎠ g⎝ x ⎠ = ⎛⎞ . ⎛ x  4⎞ ⋅⎛ x + 5⎞ ⎝ ⎠⎝ ⎠ What pieces of information do we need to gather about the graph? Where is g⎛ x ⎞ < 0 , g⎛ x ⎞ = 0 , g⎛ x ⎞ > 0 ? ⎝⎠ ⎝⎠ ⎝⎠ Let's try another. r⎛ x⎞ = ⎝⎠ 8⋅⎛ x  2⎞ ²⋅⎛ x + 1⎞ ⎝ ⎠⎝ ⎠ ⎛ x  4⎞ ⎝ ⎠
3 ⎛ x + 1⎞ ⋅⎛ x  2⎞ ⎠⎝ ⎠ , find where s⎛ x ⎞ ≤ 2 . Given s⎛ x ⎞ = ⎝ ⎝⎠ ⎝⎠ ⎛ x + 3⎞ ⋅⎛ x  6⎞ ⎝ ⎠⎝ ⎠ Section 3.7  Not on Exam 1 To find where the graphs are equal,
⎛ solve ⎜ ⎜ ⎝ ⎛ x + 1⎞ ⋅⎛ x  2⎞ ⎝ ⎠⎝ ⎠ = 2, x ⎞ ⎟ ⎟ ⎛ x + 3⎞ ⋅⎛ x  6⎞ ⎠ ⎝ ⎠⎝ ⎠ x= 161 + 5  ⎛ 161  5⎞ ⎠ or ⎜ ⎟ or x = ⎝ 2 2 x= 161 + 5  ⎛ 161  5⎞ ⎠ ⎜ ⎟ or x = ⎝ 2 2 ⎛ ⎛ x + 1⎞ ⋅⎛ x  2⎞ ⎠⎝ ⎠  2 = 0, x ⎞ solve ⎜ ⎝ ⎟ ⎜ ⎛ x + 3⎞ ⋅⎛ x  6⎞ ⎟ ⎝⎝ ⎠ ⎠⎝ ⎠ ⎛ ⎛ ⎞ 5 + 161 ⎞ 5  161 , ∞⎟ Answer: ⎜  ∞, ∪ ⎜ 3, 6⎟ ∪ ⎜ ⎜ ⎟ ⎟ 2 2 ⎝ ⎝ ⎠ ⎠ ⎜ ⎜ ⎟ ⎟ Corrected Graph: Now let's think backwards. Here is the graph, give me the function. 10 8 6 4 2 10 8 6 4 2 2 4 6 8 10
2⋅⎛ x  3⎞ ⎝ ⎠ Mistake on the handout. Vertical Asymptote at x = 2 and not x = 3. ⎛ ⎝ x + 2⎞ ⎠ 2 4 6 8 10 f ⎛ x⎞ = ⎝⎠ ...
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 Spring '06
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