3.6 rational functions...when you graph the rational functions you need to: 1. find horizontal...
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3.6 Rational Functions
Graph the rational funtion x f(x)
.1 .01 .001
1 01 001
.
.
.
1 10 100
1 10 100 State the domain and range
(-1,-1)
(1, 1) (-1, 1)
(1, -1)
This is the most basic rational function & it's negative:
These graphs both have asymptotes at x = 0 and y = 0. They are pretty easy to graph by just plotting a few points, but what do we do if we have more complicated rational functions to graph?
One of the first thing we need to do is find the x & y intercepts.
DEFINITION OF VERTICAL ASYMPTOTES
The line x=a is a vertical asymptote of the function if y approaches as x approaches a from the right or left
Example
DEFINITION OF HORIZONTAL ASYMPTOTES
The line y=b is a horizontal asymptote of the function if y approaches b as x approaches .
The next thing we need is all vertical & horizontal asymptotes.
The graph can NEVER cross a vertical asymptote,
but it CAN cross a horizontal asymptote.
Can't touch this! What did the graph say to the asymptote?
"I'm so attracted to you!"
( But only as it goes to infinity )
1. The vertical asymptotes of r are the lines x=a, where a is a zero of the denominator (after the function has been simplified).
2. (a) If n < m, then r has horizontal asymptote y=0. (b) If n=m then r has horizontal asymptote (c) If n>m, then r has no horizontal asymptote.
Let r be the rational function
FINDING ASYMPTOTES
So basically, for horizontal asymptotes THINK POWERS!If the leading term's power on the top (numerator) is smaller than the leading term's
power on the bottom (denominator), then the horizontal asymptote is _________.
If the leading term's power on the top (numerator) is equal to the leading term's
power on the bottom (denominator), then the horizontal asymptote is ___________________________________.
If the leading term's power on the top (numerator) is more than the leading term's
power on the bottom (denominator), then ___________________________.
Find the horizontal asymptotes of each function, (end behavior).
a)
b)
c)
What would the end behavior be for part c ? Since the end behavior is only determined by the first terms of the numerator and denominator, then if we only had the first terms, the function reduces to y = x. So the end behavior of this rational function is exactly like y = x, that is:
THINK "POWERS"
Find the vertical asymptotes of each function, (by setting the denominator equal to zero):a)
b)
c)
What do you think happens on the graph for the factors that cancelled when we simplified the functions? Creates a hole in the graph. To find the y-coordinate of the hole, plug in the x-value for the factor you cancelled into the reduce function. For instance, on part b, we cancelled (x- 2), so setting that equal to zero gives us x = 2. We plug 2 into
and we get the y-value of 1/5.
No V.A.
So for part (b) above:
And for part (c) above:
Find the vertical and horizontal asymptotes of
HA: powers on leading terms equal, so y = fraction of leading coefficients.
VA: set factor in the denominator equal to zero & solve.
When you graph the rational functions you need to:
1. Find horizontal asymptotes (compare powers)
2. Find the y-intercept (plug in zero for all x's)
3. Factor everything (cancel common factors - if possible)
4. Find the intercepts (numerator = 0)
5. Find vertical asymptotes (denominator = 0)
6. Find any holes (cancelled factors create holes)
7. Plot some points & sketch the graph
8. State the domain & range Step 3
must be done
before steps 4
, 5, & 6.
*
*
(Plot more points as needed.)
Multiplicity 1
Just like zeros, vertical asymptotes have multiplicity. If the factor has an odd power, (odd multiplicity), then the graph must go in opposite directions on the two sides of the asymptote. For instance, at x = 2 on this graph, the multiplicity of (x + 2) is 1, so since the graph goes to , (down), as we head toward the asymptote from the right, then the graph must go to , (up) as we head toward the asymptote from the left. (Odd multiplicity vertical asymptotes go opposite directions.)
If the multiplicity is even, then on both sides of the asymptote, the graph must go in the same direction, either both to , (both up), or to , (both down).
88
88
Domain:Range:
(No x or y intercepts)
Domain:Range:
(and state the domain and range)
Since this function reduces to y = x2, then we just graph y = x2, but make sure to put the hole in at (2, 4).
hole: (2, 4)
Domain:Range:
(and state the domain and range)
Graph , and state the domain and range.
Plot extra points as needed:
Both multiplicity 1
Domain:Range:
Graph , and state the domain and range.
(Plot more points as needed.)
Multiplicity 2
This vertical asymptote has an even multiplicity, (x1)2, so the graph must go the same direction on both sides of the asymptote x = 1. Since to the left side of the asymptote the graph heads up to , then the right side must also head up to .
8
8
Domain:Range:
Graph , and state the domain.
(Plot more points as needed.)
This vertical asymptote has an even multiplicity, (x + 5)2, so the graph must go the same direction on both sides of the asymptote x = 5. Since to the left side of the asymptote the graph heads down to , then the right side must also head down to .
8
8
Domain:
Graph , and state the domain and range.
(Plot more points as needed.)
Even multiplicity at (x 1)2, so to both sides of the asymptote x = 1, the graph heads the same direction, (in this case, both head up to ).
Odd multiplicity at (x + 1), so at the asymptote x = 1, the graphs must head in opposite directions. To the left of the asymptote, the graph heads down to , so to the right of it, the graph must head up to .
8
88
Domain:Range:
Graph , and state the domain and range.
(Plot more points as needed.)
Both multiplicity of 1
Domain:Range:
Graph , and state the domain and range.
(Plot more points as needed.)
Since the power on the leading term in the numerator is exactly 1 more than the power in the leading term of the denominator, then there is a slant asymptote. To find the equation of the slant asymptote, do long division for the rational function and the equation of the slant asymptote is the quotient. Domain:
Range:
Assignment:Section 3.6:
Problems 17 odd, 2175 odd
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