Outline Rational Functions Graphing
College Algebra & Trigonometry I 3.5 - Rational Functions and Their Graphs
Math 1100 North Carolina Central University Math & C.S. Department Hicham Qasmi -
[email protected]
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Definition Domain
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Definition Domain
Rational Functions Definition A rational function is a function of the form P(x) Q(x) where P(x) and Q(x) are polynomials with Q 6= 0. Example x +1 x 2 − 5x + 2
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Definition Domain
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Definition Domain
Domain of a Rational Function
The domain of a rational function is all the real numbers so that the denominator is never zero.
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Definition Domain
Example of Domains
Find the domain of 1 2 3
x 2 −9 x−3 x x 2 −9 x+3 x 2 +9
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Reciprocal Function
Definition The reciprocal function is the function defined by f (x) =
1 x
Its domain is all the numbers except 0.
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Limit Notation
Symbol x → a+ x → a− x →∞ x → −∞
Meaning x approaches a from the right x approaches a from the left x approaches infinity x approaches negative infinity
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Vertical Asymptotes
Definition The line x = a is a vertical asymptote of the graph of f if f (x) increases or decreases without bound as x approaches a.
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Finding Vertical Asymptotes
Theorem P(x) If f (x) = Q(x) is a rational function and P(x) and Q(x) have no common factors, if a is a zero of Q(x), then x = a is a vertical asymptote.
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Vertical Asymptotes Example
Example Find the vertical asymptotes of 1 2 3
x x 2 −9 x+3 x 2 −9 x+3 x 2 +9
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Horizontal Asymptotes
Definition The line y = b is a horizontal asymptote of the graph of f if f (x) approaches b as x increases or decreases without bound(approaches ∞ or −∞).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Horizontal Asymptote
Example 1 x
−→ 0 when x −→ ∞, so the line y = 0 is a horizontal asymptote of x1 .
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Finding Horizontal Asymptotes Theorem Let f be a rational function given by f (x) =
ap x p + . . . + a1 x + a0 b q x q + · · · + b 1 x + b0
with an 6= 0, bm 6= 0
The degree of the numerator is p and the degree of the denominator is q. 1
If p < q, then the x-axis (y = 0) is the horizontal asymptote for f .
2
If p = q, then the line y = asymptote for f .
3
If p > q, then there is no horizontal asymptote. Math 1100
ap bq
is the horizontal
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Finding Horizontal Asymptotes Theorem Let f be a rational function given by f (x) =
ap x p + . . . + a1 x + a0 b q x q + · · · + b 1 x + b0
with an 6= 0, bm 6= 0
The degree of the numerator is p and the degree of the denominator is q. 1
If p < q, then the x-axis (y = 0) is the horizontal asymptote for f .
2
If p = q, then the line y = asymptote for f .
3
If p > q, then there is no horizontal asymptote. Math 1100
ap bq
is the horizontal
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Finding Horizontal Asymptotes Theorem Let f be a rational function given by f (x) =
ap x p + . . . + a1 x + a0 b q x q + · · · + b 1 x + b0
with an 6= 0, bm 6= 0
The degree of the numerator is p and the degree of the denominator is q. 1
If p < q, then the x-axis (y = 0) is the horizontal asymptote for f .
2
If p = q, then the line y = asymptote for f .
3
If p > q, then there is no horizontal asymptote. Math 1100
ap bq
is the horizontal
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Finding Horizontal Asymptotes Theorem Let f be a rational function given by f (x) =
ap x p + . . . + a1 x + a0 b q x q + · · · + b 1 x + b0
with an 6= 0, bm 6= 0
The degree of the numerator is p and the degree of the denominator is q. 1
If p < q, then the x-axis (y = 0) is the horizontal asymptote for f .
2
If p = q, then the line y = asymptote for f .
3
If p > q, then there is no horizontal asymptote. Math 1100
ap bq
is the horizontal
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Finding Horizontal Asymptotes Theorem Let f be a rational function given by f (x) =
ap x p + . . . + a1 x + a0 b q x q + · · · + b 1 x + b0
with an 6= 0, bm 6= 0
The degree of the numerator is p and the degree of the denominator is q. 1
If p < q, then the x-axis (y = 0) is the horizontal asymptote for f .
2
If p = q, then the line y = asymptote for f .
3
If p > q, then there is no horizontal asymptote. Math 1100
ap bq
is the horizontal
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Horizontal Asymptotes Examples
Example Find the horizontal asymptote for 1
2
3
4x 2x 2 +1 4x 2 2x 2 +1 4x 3 2x 2 +1
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Transformation on Rational Functions Graphing a rational function
Example Use the graph of f (x) =
1 x
to graph
g(x) =
1 +1 (x − 2)2
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline 1
Rational Functions Definition Domain
2
Graphing Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Strategy for Graphing a Rational Function The following strategy can be used to graph f (x) = P(x)/Q(x) 1
Determine the symmetry of the graph: f (−x) = f (x), even f (−x) = −f (x), odd
2
Find the y -intercept.
3
Find the x-intercept(s).
4
Find the vertical asymptote(s)
5
Find the horizontal asymptote
6
Plot at least one point between and beyond each x-intercept and vertical asymptote.
7
Finish by graphing the function between and beyond the vertical asymptotes(use all the info obtained above).
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Graphing a Rational Function Example
Example Graph f (x) =
Math 1100
2x x −1
College Algebra & Trigonometry I
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline Rational Functions Graphing
Graphing a Rational Function Example 5 4
•
3 • 2 •
y =2
• 1
0• −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
x =1 •
−3 −4 −5
Math 1100
College Algebra & Trigonometry I
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Outline Rational Functions Graphing
Graphing a Rational Function Example 5 4
•
3 • 2 •
y =2
• 1
0• −5 −4 −3 −2 −1 0 −1 −2
1
2
3
4
5
x =1 •
−3 −4 −5
Math 1100
College Algebra & Trigonometry I
Outline Rational Functions Graphing
Reciprocal Function Limit Notation Vertical and Horizontal Asymptotes Transformation on Rational Functions Strategy for Graphing
Graphing a Rational Function Example
Example Graph f (x) =
Math 1100
3x 2 x2 − 4
College Algebra & Trigonometry I
