3.5 Rational Functions and Their Graphs.notebook
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3.5: Rational Functions and Their Graphs Date: 10/25
Rational Functions are quotients of polynomial functions. This means that a rational function can be expressed as:
Where p and q are polynomial functionsand
The domain of a rational function is the set of all real numbers except the x‐values that make the denominator zero.
3.5 Rational Functions and Their Graphs.notebook
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Ex 1: Find the domain of each rational function. (Use interval notation)
a) b) c)
3.5 Rational Functions and Their Graphs.notebook
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Another basic rational function is
3.5 Rational Functions and Their Graphs.notebook
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Vertical Asymptote: The line x = a of the graph of a function f if f(x) increases or decreases without bound as x approaches a.
As As
As As
3.5 Rational Functions and Their Graphs.notebook
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Locating a Vertical Asymptote: If is a rational function in which p(x) and
q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.
Ex 2: Find the vertical asymptotes, if any, of the graph of each rational function.
a) b) c)
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Note: A value where the denominator of a rational function is zero does not necessarily result in a vertical asymptote. There is a hole in the graph at x = a, and not a vertical asymptote, under the following conditions: the value a causes the denominator to be zero, but there is a reduced form of the function’s equation in which a does not cause the denominator to be zero. (The factor x ‐ a cancels out when the rational function is simplified).
3.5 Rational Functions and Their Graphs.notebook
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Go back to Ex 2. Are there any functions that have holes in the graph? Identify where this occurs.
3.5 Rational Functions and Their Graphs.notebook
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Horizontal Asymptote: The line y = b of the graph of a function f if f(x) approaches b as x increases or decreases without bound.
As
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Locating Horizontal Asymptotes:
Let f be a rational function given by:
The degree of the numerator is: ________The degree of the denominator is: _______
1.
2.
3.
Examples:
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Ex 3: Find the horizontal asymptote, if any, of the graph of each rational function:
a) b) c)
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Using Transformations to Graph Rational Functions:
Ex 4: Use the graph of to graph
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Ex 5: Use the graph of to graph