1. 25. Rational Functions
Analyzing and sketching the graph of a rational function

2. where N and D are both polynomials.
Rational Functions
A function of the form
f(x) = N(x)/D(x),
where N and D are both polynomials.
Examples:

3. What we will be doing
Finding some characteristics of rational functions that will help us to be able to graph the function
Graphing the function (we will use the calculator to help us and check ourselves but you must show work that leads to graph)
Using graph to find some other characteristics

4. Characteristics we will find to help graph
Horizontal Asymptote (HA)
Vertical Asymptote
Holes
Points of Discontinuity – VA and holes
y- intercept
x- intercept(s)

5. Characteristics found after graphing
Domain – all x values, VA and holes will be excluded
Range – all y values, HA will be excluded
Extrema – y values that are a minimum or maximum – they can be relative or absolute
End behavior – what happens to the y-values on left and right sides, will match the HA
Intervals of Increase (II)– where graph is going up
Intervals of Decrease (ID)– where graph is going down
Rate of change - find y values for 2 given x values, then use slope formula to calculate

6. Horizontal Asymptotes
Compare the largest exponents in numerator and denominator
1) Bigger On Bottom y=0
2) Exponents Are The Same;
Divide the Coefficients
3) Bigger On Top – Slant (see next page)

7. SLANT ASYMPTOTES
If the higher exponent is on top, there is a SLANT asymptote. We will not learn slant asymptotes so we will write “NONE”

8. FINDING HA - examples

9. Continuity A function is CONTINUOUS if you can draw the graph
without lifting your pencil.
A POINT OF DISCONTINUITY occurs when
there is a break in the graph.
There are 2 types of discontinuity we will look at

10. Hole: a single point at which the graph has no value
DISCONTINUITIES
Asymptote: a line that the graph approaches more and more closely but will never touch.
Hole: a single point at which the graph has no value

11. To find VA and HOLES Factor the numerator and denominator
Simplify any factors that are in common
Anything that is left in the denominator, set equal to 0 – this is a VA
Whatever you were able to simplify (might be nothing), set equal to 0 – this is a hole

12. HOLES VS VA - Examples
VA x = Hole x = -5
VA x = 6 Hole x = -5/3

13. FINDING THE X-INTERCEPTS
After factoring, set what is left in the numerator equal to 0 and solve for x (could be none)
FINDING THE Y-INTERCEPTS
Plug in 0 for x and solve. Answer must be written y =

14. Find the x and y intercepts2.
x-intercepts:
y-intercept:
x-intercepts:
y-intercept:
x-intercepts:
y-intercepts
x-intercepts:
y-intercept:

15. Find the HA, VA, holes, P of D, x-intercepts, y-intercepts
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:

16. Summary HA – find before factoring by comparing biggest exponent
VA – find after factoring by setting denominator = 0
Holes – find after factoring by setting canceled out factor = 0
POD – VA and holes
X intercepts – find after factoring by setting numerator = 0
Y intercepts – find before factoring by plugging in 0 for x