rational functions a rational function is a function of the form: where p and q are polynomials
TRANSCRIPT
RATIONAL
FUNCTIONSA rational function is a function of the form:
xqxp
xR where p and q are polynomials
xqxp
xR What would the domain of a rational function be?
We’d need to make sure the denominator 0
x
xxR
3
5 2
Find the domain. 3: xx
22
3
xx
xxH 2,2: xxx
45
12
xx
xxF
If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values.
014 xx 1,4: xxx
The graph of looks like this: 2
1
xxf
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
If you choose x values close to 0, the graph gets close to the asymptote, but never touches it.
Let’s consider the graph x
xf1
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.
10
1011
10
1
f
100
10011
100
1
f
10
101
1
10
1
f 100
1001
1
100
1
f
The closer to 0 you get for x (from positive
direction), the larger the function value will be Try some negatives
Does the function have an x intercept? x
xf1
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)
x
10
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
Graph x
xQ1
3
This is just the reciprocal function transformed. We can trade the terms places to make it easier to see this.
31
x
vertical translation,
moved up 3
x
xf1
x
xQ1
3
The vertical asymptote remains the same because in either function, x ≠ 0
The horizontal asymptote will move up 3 like the graph does.
Finding AsymptotesVER
TIC
AL A
SYM
PTO
TE
S
There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0
43
522
2
xx
xxxR
Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.
014 xx
So there are vertical asymptotes at x = 4 and x = -1.
Hole (in the graph)
• If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote.
• The exact point of the hole can be found by plugging b into the function after it has been simplified.
• Huh???? Let’s look at an example or two.
Find the domain and identify vertical asymptotes & holes.
2
1( )
2 3
xf x
x x
Find the domain and identify vertical asymptotes & holes.
2( )
4
xf x
x
Find the domain and identify vertical asymptotes & holes.
2
5( )
2 3
xf x
x x
Find the domain and identify vertical asymptotes & holes.
2
2
3 2( )
2
x xf x
x x
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.
We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.
43
522
xx
xxR
degree of bottom = 2
HORIZONTAL ASYMPTOTES
degree of top = 1
1
1 < 2
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 bottom = 2
HORIZONTAL ASYMPTOTES
degree of top = 2
The leading coefficient is the number in front of the highest powered x term.
horizontal asymptote at:
1
2
43
5422
2
xx
xxxR
1
2y
43
5322
23
xx
xxxxR
If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique (a/k/a slant) one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder.
degree of bottom = 2
OBLIQUE (SLANT) ASYMPTOTES
degree of top = 3
532 23 xxx432 xx
remainder a 5x
Oblique (slant) asymptote at y = x + 5
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 oblique asymptotes, compare the degrees of the numerator and denominator.
1. If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0)
2. If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom
3. If the degree of the top > the bottom, oblique (slant) asymptote found by long division.