Download Graphing and Analyzing Rational Functions and more Exams Algebra in PDF only on Docsity!
179
L17 Graphing Rational Functions; Rational Inequalities
Graphing Rational Functions
Factor both the numerator and denominator.
Write the domain.
Reduce the fraction to lowest terms, if possible.
Give the vertical asymptotes and holes.
Find the
(^) x -intercept(s) and
(^) y -intercept.
Find the horizontal or oblique asymptote, if any.
oblique asymptote.Determine if the graph crosses its horizontal or
Test for symmetry.
Plot all asymptotes, holes, intercepts, and points of
10.crossing of horizontal/oblique asymptote. (^) Using the end behavior and multiplicities of zeroes
11.intercepts/asymptotes.sign of the function on each interval between theof both the numerator and denominator, determine the (^) Plot a few more points where it is needed.
(^) Graph the function.
180
Example: Graph the rational function
(^3)
2
2
x
x
x
R x
x x
Horizontal or Oblique Asymptote: y -intercept: x -intercept(s):VA(s): Hole(s): Domain:
181
Symmetry: Crossing of the horizontal or oblique asymptote:
182
Example: Construct a rational function from its graph.
185
population of the manatees is given by50 Manatees are taken to a river sanctuary. The Horizontal Asymptotes and Limiting Size of Population:
t
N t
t
t
where
(^) t is time in years.
Find the population after
t (^) =
years; after 100 years.
increases?What is the limiting size of the population as time
186
1. Get all terms on the left side of the inequality with a 0 Solving Rational Inequalities
single fraction. Write the domain. Reduce the fraction.on the right side and simplify the left-hand side into a
- Find all real zeros of the numerator and denominator. Determine their multiplicities.
- Divide a real line into intervals using the zeros found as endpoints. Label an endpoint asin Step 2 and the numbers that are not in the domain
(^) if it is to be
included in the answer and label it as
D
(^) if it is not.
Note:
Zeros of the denominator are never included!
included if and only if the inequality is non strict (Zeros of the numerator which are in the domain are
- Use the end behavior of the polynomials in the fraction on the right-most interval (whennumerator and denominator to find the sign of the
(^) x (^) → +∞
- Set the signs on each other interval by moving from sign depending on the multiplicity.the right to the left and changing/not changing the
- Select the intervals with the desired sign of the fraction according to the inequality in Step 1.
187
Important:
(^) Never multiply or divide both sides of an
and varies its sign depending on the variable.inequality by an expression that contains a variable
Example: Solve
(^32)
x^ x +
188
Example:
Solve
2
3
x
x
x
x
x
x
