1. RATIONAL FUNCTIONS A rational function is a function of the form:
where p and q are polynomials

2. What would the domain 정의역 of a rational function be?
We’d need to make sure the denominator  0
Find the domain.
Set the denominator = 0 and factor to find “illegal” values.

3. Asymptotes
Places on the graph the function will approach, but will never touch.

4. Let’s consider the graph
We recognize this function as the reciprocal function from our “library” of functions.
Can you see the vertical asymptote?
Since x  0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0

5. Let’s consider the graph
We recognize this function as the reciprocal function from our “library” of functions.
Can you see the vertical asymptote?
Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0.
The closer to 0 you get for x (from positive direction), the larger the function value will be
Try some negatives

6. Does the function have an x intercept?
There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote.
A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0)
A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left

7. Finding Asymptotes VERTICAL ASYMPTOTES
There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0
So there are vertical asymptotes at x = 4 and x = -1.
VERTICAL ASYMPTOTES
Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.

8. HORIZONTAL ASYMPTOTES
We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.
1 < 2
degree of top = 1
If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote.
If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0. 1
degree of bottom = 2

9. HORIZONTAL ASYMPTOTES
The leading coefficient is the number in front of the highest powered x term.
If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at:
y = leading coefficient of top
leading coefficient of bottom
degree of top = 2 1
degree of bottom = 2
horizontal asymptote at:

10. SLANT (OBLIQUE )ASYMPTOTES
If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an slant one. The equation is found by doing long division and the quotient is the equation of the slant asymptote ignoring the remainder.
degree of top = 3
degree of bottom = 2
Oblique asymptote at y = x + 5

11. SUMMARY OF HOW TO FIND ASYMPTOTES
Vertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve.
To determine horizontal or slant asymptotes, compare the degrees of the numerator and denominator.
If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0)
If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom
If the degree of the top > the bottom, the slant asymptote is found by long division.

12. Continuous functions are predictable…
If a function is continuous at every value in an interval, then we say that the function is continuous in that interval.  And if a function is continuous in any interval, then we simply call it a continuous function.
No breaks in the graph
No holes
No jumps

13. A function such as g(x)=
A function such as g(x)= has a discontinuity at x=3 because the denominator is zero there

14. The graph can have a cusp (an abrupt change in direction) at x=c and still be continuous there

15. Discontinuity vs. Holes
A graph is discontinuous where the denominator equals 0. Where the x-value that makes the denominator 0 also makes the numerator 0, then it is also called a HOLE. Denominator = 0 –> discontinuity Both numerator and denominator are 0 is also called a Hole

16. Steps to graphing rational functions
Find the y-intercept.
Find the x-intercepts.
Find vertical asymptote(s).
Find horizontal asymptote.
Find any holes in the function.
Make a T-chart: choose x-values on either side & between all vertical asymptotes.
Graph asymptotes, pts., and connect with curves.

17. Examples Graph y = 6 x2 + 3 No x-intercepts No vertical asymptote2 -1
1.5 1 -2
6/7
No x-intercepts
No vertical asymptote
H.A.: y = 0

18. Let's try another
vertical asymptote from this factor only since other factor cancelled.
But notice that the top of the fraction will factor and the fraction can then be reduced.
We don’t have a vertical asymptote at x = -3, but it is still an excluded value NOT in the domain.
Find the domain. Excluded values are where your vertical asymptotes are.

19. We'll graph the reduced fraction but we must keep in mind that x  - 32 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
So let’s plot the y intercept which is (0, - 1/3)
Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x

20. x + 1 = 0 when x = -1 so there is an x intercept at the point (-1, 0)2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
If the numerator of a fraction = 0 then the whole fraction = 0 since 0 over anything = 0
x + 1 = 0 when x = -1 so there is an x intercept at the point (-1, 0)
Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving.

21. Not the original and not negative of function so neither even nor odd.2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
Not the original and not negative of function so neither even nor odd.
Test for symmetry by putting –x in for x. (remember even, odd test)

22. 1 1 degree of the top = 1 degree of the bottom = 12 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 1 1
degree of the bottom = 1
If the degree of the top equals the degree of the bottom then there is a horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom.
Find horizontal or oblique asymptote by comparing degrees

23. We already have some points on the left side of the vertical asymptote so we can see where the function goes there 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 x
S(x) 4 5 6
2.3
Let's choose a couple of x's on the right side of the vertical asymptote.
Find some points on either side of each vertical asymptote

24. Pass through the point and head towards asymptotes
Pass through the points and head towards asymptotes
There should be a piece of the graph on each side of the vertical asymptote. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
REMEMBER that x  -3 so find the point on the graph where x is -3 and make a "hole" there since it is an excluded value.
Connect points and head towards asymptotes.

25. Your turn
Practice finding asymptotes, holes and try to graph