1. Chapter 2
2-7 rational functions

2. SAT Problem of the day
Point A has coordinates (-4,3), and the midpoint of AB is the point (1,-1). What are the coordinates of B.
A)(-3,4)
B)(-4,5)
C)(4,-5)
D)(5,-4)
E)(6,-5)

3. Solution
Right Answer: E

4. Objectives Analyze and sketch graphs of rational functions
Use the slant asymptote to graph

5. How we graph rational functions
To graph a rational function, you find the asymptotes and the intercepts, plot a few points, and then sketch in the graph. Once you get the swing of things, rational functions are actually fairly simple to graph. Let's work through a few examples.

6. Example#1 Graph the following: Solution:
First I'll find the vertical asymptotes, if any, for this rational function. Since I can't graph where the function doesn't exist, and since the function won't exist where there would be a zero in the denominator, I'll set the denominator equal to zero to find any forbidden points:
x – 1 = 0 x = 1

7. Solution
So I can't have x = 1, and therefore I have a vertical asymptote there.
I'll dash this in on my graph:   

8. solution
Next I'll find the horizontal or slant asymptote. Since the numerator and denominator have the same degree (they're both linear), the asymptote will be horizontal, not slant, and the horizontal asymptote will be the result of dividing the leading coefficients:
y = 2/1 = 2
I'll dash this in, too:   

9. solution Next, I'll find any x- or y-intercepts.
x = 0:  y = (0 + 5)/(0 – 1) = 5/–1 = –5 y = 0:  0 = (2x + 5)/(x – 1)               0 = 2x + 5         –5 = 2x      –2.5= x
Then the intercepts are at (0, –5) and (–2.5, 0). I'll sketch these in:   
     –2.5 = x

10. solution
Now I'll pick a few more x-values, compute the corresponding y-values, and plot a few more points.
Now I'll plot these points:   

11. solution
And now I can connect the dots

12. solution

13. Example#2
Graph the following:

14. Example#3
Graph the following:

15. What is the slant asymptote
Slanted or Oblique asymptotes occur in rational functions where the degree of the numerator is higher than the degree of the denominator.
. The equation for the slant asymptote is the polynomial part of the rational that you get after doing the long division

16. Example#4 Find the slant asymptote of the following function:
slant asymptote:  y = x + 5

17. Example#5 Find the slant asymptote of the following function:
slant asymptote: y = –2x – 4

18. Student guided practice
Work on problems on your worksheet graphing rational functions

19. homework
Do problems 17-20, on your book page 157 and 158.

20. Closure Today we learned about graphing rational functions
Next class we are going to learned about exponential functions