Download Steps to Graph a Rational Function

Graphs of Rational Functions
(Section 2-7)
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Steps to Graph a Rational Function
1. Simplify f if possible. Any restrictions on the domain of f not in the simplified
function should be listed.
2. Find and plot the y-intercept (if any) by evaluating f(0).
3. Find and plot the zeros by finding the values that make the numerator equal zero.
4. Find and sketch any vertical asymptotes by finding the values that make the
denominator equal zero.
•
Plot any holes using an open circle.
5. Find and sketch any other asymptotes.
6. Plot at least one point between and one point beyond each x-intercept and vertical
asymptote.
7. Use smooth curves to complete the graph between and beyond the vertical
asymptotes, excluding any points where f is not defined.
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Use a graphing utility to graph f(x) = 3/x and the function g in the same viewing window.
Describe the relationship between the two graphs.
Example 1
g(x) = f(x) + 2
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Use a graphing utility to graph f(x) = 3/x and the function g in the same viewing window.
Describe the relationship between the two graphs.
Example 2
g(x) = f(x + 4)
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Use a graphing utility to graph f(x) = 3/x and the function g in the same viewing window.
Describe the relationship between the two graphs.
Example 3
g(x) = -f(x)
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Use a graphing utility to graph f(x) = 3/x and the function g in the same viewing window.
Describe the relationship between the two graphs.
Example 4
g(x) = 2 f(x) + 1
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Sketch the graph of the rational function by hand. As sketching aides, check for
intercepts, vertical asymptotes, horizontal asymptotes and holes. (Same as #9-25)
Example 5
g ( x) 
3
x2
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Sketch the graph of the rational function by hand. As sketching aides, check for
intercepts, vertical asymptotes , horizontal asymptotes and holes. (Same as #9-25)
Example 6
f ( x) 
2x 1
x
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Sketch the graph of the rational function by hand. As sketching aides, check for
intercepts, vertical asymptotes , horizontal asymptotes and holes. (Same as #9-25)
Example 7
f ( x) 
x
x2  x  2
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Sketch the graph of the rational function by hand. As sketching aides, check for
intercepts, vertical asymptotes , horizontal asymptotes and holes. (Same as #9-25)
Example 8
x2  9
f ( x)  2
x  2x  3
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
HW #68 pg161 (1-41 odd)
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Slant Asymptotes
If the degree of the numerator is exactly one more than the degree of the
denominator there is a slant asymptote.
To find the equation of a slant asymptote use long division to divide the numerator by
the denominator. The result (quotient) of the long division is the equation of the slant
asymptote, excluding the remainder.
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Find the slant asymptotes.
Example 9
x2  x
f ( x) 
x 1
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Sketch the graph of the rational function by hand. As sketching aides, check for intercepts,
vertical asymptotes and slant asymptotes.
Example 10
x3
f ( x)  2
x 1
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Sketch the graph of the rational function by hand. As sketching aides, check for intercepts,
vertical asymptotes and slant asymptotes.
Example 11
1 x2
f ( x) 
x
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Find all vertical asymptotes, horizontal asymptotes, slant asymptotes and holes in the
graph of the function.
3
2
Example 12 f ( x)  2 x  x  8 x  4
2
x  3x  2
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Use a graphing utility to graph the function and determine any x-intercepts. Set y = 0 and
solve the resulting equation to confirm your result.
Example 14
y
2
3

x 1 x
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
Use a graphing utility to graph the function and determine any x-intercepts. Set y = 0 and
solve the resulting equation to confirm your result.
Example 15
y  x
9
x
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.
HW #69 pg 161 – 162 (43-71 odd)
Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes?
Students will write a summary describing the different parts of a graph of the rational function.