Transcript Rational Functions

25. Rational Functions
Rational Functions
A function of the form
f(x) = N(x)/D(x),
where N and D are both polynomials.
Examples:
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
Characteristics we will
find to help graph
1.
2.
3.
4.
5.
6.
Horizontal Asymptote (HA)
Vertical Asymptote
Holes
Points of Discontinuity – VA and holes
y- intercept
x- intercept(s)
Characteristics found
after graphing
7.
8.
9.
10.
11.
12.
13.
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
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)
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”
FINDING HA - examples
5
y
2x  7
7x
y 2
3x  4 x
3x  5 x  7
y 2
6 x  4 x  11
2
x  3x  10
y
x3
2
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
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
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
HOLES VS VA Examples
x  3x  10
y 2
x  8 x  15
2
•
VA x = -3
Hole x = -5
6 x  2 x  20
y 2
3x  13x  30
2
•
x  5  x  2 

y
 x  5 x  3
VA x = 6 Hole x = -5/3
2 x  4  3x  5

y
 3x  5 x  6
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=
Find the x and y intercepts
1.
2
y 2
x 2
x-intercepts:
y-intercept:
2
2x
y 2
x 9
x-intercepts:
y-intercept:
x  5x  6
y
x3
x  3x  10
4. y  2
x  8 x  15
x-intercepts:
y-intercepts
x-intercepts:
y-intercept:
2
3.
2.
2
Find the HA, VA, holes, P of D, x-intercepts, y-intercepts
1.
2
y
x2
2.
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
x
3. y  2
x 4
x  2x  3
y
2
x 9
2
2
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
4
y 2
x 4
4.
HA:
VA:
Holes:
P of D:
x-intercepts:
y-intercepts:
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