1. Algebra 1
Section 13.8

2. Definition
A rational function is a function expressed as a ratio of polynomials whose denominator is not zero.

3. Example 1 Graph f(x) = . x ≠ 0; the graph is undefined for x = 0.2 x
x ≠ 0; the graph is undefined for x = 0.
Make a table of ordered pairs.

4. Example 1
x y -4 -½ -2 -1 -1 -2 -½ -4
undefined 4 1 2 2 1 4

5. Example 1 Graph f(x) = . This graph is called a hyperbola.2 x
This graph is called a hyperbola.
The ends of each branch approach but never touch the x- and y-axes.

6. Example 1
Graph f(x) = 2 x
Therefore, the x-axis is the horizontal asymptote and the y- axis is the vertical asymptote.
The center of the hyperbola is at (0, 0).

7. Example 2
Graph f(x) = 2 x
Since the function is undefined when x = 0, x = 0 is the equation of a vertical asymptote.
Make a table of ordered pairs.

8. Example 2
x y -4 2½ -2 2 -1 1 -½ -1
undefined 7 1 5 2 4 4 3½

9. Example 2
Graph f(x) = 2 x
This graph is the result of translating the graph of f(x) = three units up. 2 x
The center moves up to (0, 3).

10. Example 2
Graph f(x) = 2 x
The horizontal asymptote moves up to y = 3.
The vertical asymptote is still the y-axis.

11. Graphing Rational Functionsx
In general, the graph of
f(x) = k is the result of translating the graph of f(x) =
k units up if k is positive and k units down if k is negative.

12. Example 3
Graph g(x) = -6 x
x = 0 is the equation of a vertical asymptote.
Make a table of ordered pairs.
The graph is in the upper left and lower right quadrants.

13. Example 3
Graph g(x) = -6 x
The horizontal asymptote is the x-axis, or y = 0.
The center is at (0, 0).

14. Example 4
Graph h(x) = -6
x – 2
x = 2 is the equation of a vertical asymptote.
Make a table of ordered pairs.
Graph the function.

15. Example 4
Graph h(x) = -6
x – 2
The graph of h(x) is the result of translating the graph of g(x) [Example 3] 2 units right.

16. Graphing Rational Functions
The graph of a function of the form y = k is a translation of the base function y = It is a hyperbola with... a x
x – h
...a horizontal asymptote of y = k.
...its center at (h, k).
...a vertical asymptote of x = h.

17. Example 5 Graph f(x) = – 1. (h, k) = (-3, -1)2
x + 3
(h, k) = (-3, -1)
vertical asymptote: x = -3
horizontal asymptote: y = -1
Draw the asymptotes and label the center.

18. Example 5
Graph f(x) = – 1. 2
x + 3
Make a table of ordered pairs using several convenient points on either side of the hyperbola’s center.
Graph the function.

19. Example 5 The graph of f(x) = – 12
The graph of f(x) = – 1
is the result of translating the graph of f(x) = 2/x three units left and one unit down.
This is similar to the translations of other types of functions studied earlier.
x + 3

20. Homework: pp