1. x -4 -2 -1 -½ ½ 1 2 4 8-4 LEARNING GOALS
Graph rational functions.
Transform rational functions by changing parameters.
Identify vertical and horizontal asymptotes, and holes
The parent function for the family of rational functions is:
Its graph is called a _______________. They are characterized by two separate branches and two asymptotes.
Domain:
Range: x -4 -2 -1 -½ 1 2 4
A _____________________ function is a function whose graph has one or more gaps or breaks.
Example: Hyperbolas
A _____________________ function is a function whose graph has no gaps or breaks.
Examples: Lines, Parabolas, Logarithmic Curves, Exponential Functions
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2. Example 1: Transforming Rational Functions
Using the graph of as a guide, describe the transformation and graph each function.
A. B.
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3. Example 2: Determining Properties of Hyperbolas
Using the graph of as a guide, describe the transformation and graph each function. C.
Example 2: Determining Properties of Hyperbolas
A. Identify the asymptotes, domain, and range of the function
Sketch the graph:
Vertical asymptote:
Domain:
Horizontal asymptote:
Range:
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4. ZEROS B. Identify the asymptotes, domain, and range of the function
Vertical asymptote:
Sketch the graph:
Domain:
Horizontal asymptote:
Range:
C. Identify the asymptotes, domain, and range of the function
Vertical asymptote:
Sketch the graph:
Domain:
Horizontal asymptote:
Range:
ZEROS
What is a “zero”?
What does it look like graphically?
What has to be true about a fraction to make it = 0
Example 3: Finding Zeros
Identify the zeros of f(x). A. B.
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5. Example 3 Continued: Finding Asymptotes
Vertical asymptotes are written as: X = _____
They are found by factoring the denominator and setting it equal to zero.
Horizontal asymptotes are written as: Y = ______
They are found by looking at the degree of the numerator and denominator using the chart below.
Degree of numerator & denominator
N > D
N < D
N = D
Type of horizontal asymptote
None
y = 0
y =
Identify both the vertical and horizontal asymptotes of the function. A. B. C. D.
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6. Example 4: Graphing Rational Functions with Holes
When a functions numerator = ____ and denominator = ____ for the same x value the graph may have a ____________.
HOLE:
NO HOLE:
Example 4: Graphing Rational Functions with Holes
Identify holes in the graph of f(x). Then graph.
A B.
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