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Rational Functions
In this chapter, youâll learn what a rational function is, and youâll learn how to sketch the graph of a rational function.
Rational functions
A rational function is a fraction of polynomials. That is, if p(x) and q(x) are polynomials, then p(x) q(x)
is a rational function. The numerator is p(x) and the denominator is q(x).
Examples.
3(x 5) (x 1)
3 1 = 2x^
3
The last example is both a polynomial and a rational function. In a similar way, any polynomial is a rational function.
In this class, from this point on, most of the rational functions that weâll see will have both their numerators and their denominators completely factored. We will also only see examples where the numerator and the denominator have no common factors. (If they did have a common factor, we could just cancel them.)
Implied domains
The implied domain of a rational function is the set of all real numbers except for the roots of the denominator. Thatâs because it doesnât make sense to divide by 0.
Example. The implied domain of 7(x 2)(x 2 + 1) 8(x 4)(x 6) is the set R { 4 , 6 }.
Vertical asymptotes
To graph a rational function, begin by marking every number on the x-axis that is a root of the denominator. (The denominator might not have any roots.) Draw a vertical dashed line through these points. These vertical lines are called vertical asymptotes. The graph of the rational function will âclimb upâ or âslide downâ the sides of a vertical asymptote.
Examples. For the rational function (^) x^1 , 0 is the only root of the denominator, so the y-axis is the vertical asymptote. Notice that the graph of (^1) x climbs up the right side of the y-axis and slides down the left side of the y-axis.
The rational function 7(x 2)(x 2 + 1) 8(x 4)(x 6)
has vertical asymptotes at x = 4 and at x = 6.
Example. The implied domain of
7(x 2)(x^2 + 1) 8(x 4)(x 6)
is the set R { 4 , 6 }.
Vertical asymptotes
To graph a rational function, begin by marking every number on the x-axis that is a root of the denominator. (The denominator might not have any roots.) Draw a vertical dashed line through these points. These vertical lines are called vertical asymptotes. The graph of the rational function will âclimb upâ or âslide downâ the sides of a vertical asymptote.
Examples. For the rational function (^1) x , 0 is the only root of the denominator, so the y-axis is the vertical asymptote. Notice that the graph of (^) x^1 climbs up the right side of the y-axis and slides down the left side of the y-axis.
The rational function 7(x 2)(x^2 + 1) 8(x 4)(x 6)
has vertical asymptotes at x = 4 and at x = 6.
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Notice that 7, (3 4), and (3 6) are negative, while 8, (3 2), and (3 2 + 1) are positive. If you are multiplying and dividing a collection of numbers that arenât equal to 0, just count how many negative numbers there are. If there is an even number of negatives, the result will be positive. If there is an odd number of negatives, the result will be negative. In the previous paragraph, there are three negative numbers â 7, (3 4), and (3 6) â so r(3) < 0. The number 5 is a number that is in between 4 and 6, and
r(5) =
Far right and far left
Let ax n^ be the leading term of p(x) and let bx m^ be the leading term of q(x). Recall that far to the right and left, p(x) looks like its leading term, ax n^. And far to the right and left, q(x) looks like its leading term, bx m^. It follows that the far right and left portion of the graph of,
p(x) q(x)
looks like ax n bx m
and this is a function that we know how to graph.
Example. The leading term of 7(x 2)(x 2 + 1) is 7 x 3 , and the leading term of 8(x 4)(x 6) is 8x 2. Therefore, the graph of
r(x) =
7(x 2)(x 2 + 1) 8(x 4)(x 6)
looks like the graph of 7 x 3 8 x 2
x
on the far left and far right part of its graph.
Putting the graph together
To graph a rational function p(x) q(x) mark its vertical asymptotes (if any). Mark its x-intercepts (if any). Deter- mine whether the function is positive or negative in between x-intercepts and vertical asymptotes. Replace p(x) with its leading term, replace q(x) with its leading term, and then graph the resulting fraction of leading terms to the right and left of everything youâve drawn so far in your graph. Now draw a reasonable looking graph that fits with everything youâve drawn so far, remembering that the graph has to climb up or slide down the sides of vertical asymptotes, and that the graph can only touch the x-axis at the x-intercepts that you already marked.
Example. Letâs graph
r(x) =
7(x 2)(x 2 + 1) 8(x 4)(x 6)
First we mark its its vertical asymptotes, which are at x = 4 and x = 6, and its x-intercept, which is at x = 2.
Putting the graph together
To graph a rational function p(x) q(x) mark its vertical asymptotes (if any). Mark its x-intercepts (if any). Deter- mine whether the function is positive or negative in between x-intercepts and vertical asymptotes. Replace p(x) with its leading term, replace q(x) with its leading term, and then graph the resulting fraction of leading terms to the right and left of everything youâve drawn so far in your graph. Now draw a reasonable looking graph that fits with everything youâve drawn so far, remembering that the graph has to climb up or slide down the sides of vertical asymptotes, and that the graph can only touch the x-axis at the x-intercepts that you already marked.
Example. Letâs graph
r(x) =
7(x 2)(x^2 + 1) 8(x 4)(x 6)
First we mark its its vertical asymptotes, which are at x = 4 and x = 6, and its x-intercept, which is at x = 2.
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a 61 I I I I I I a I I I I I
Now we connect what weâve drawn so far, making sure our graph climbs up or slides down the vertical asymptotes, and that it only touches the x-axis at the previously labelled x-intercept.
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Now we connect what weâve drawn so far, making sure our graph climbs up or slides down the vertical asymptotes, and that it only touches the x-axis at the previously labelled x-intercept.
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a
Iâ
rfr)
Exercises
For #1-3, use that 4x 2 4 = 4(x 1)(x + 1), x 3 3 x 2 + 4 = (x + 1)(x 2) 2 , and 2x 4 = 2(x 2) to match each of the three numbered rational functions on the left with its simplified lettered form on the right.
1.) 4 x^
(^2) 4 x 3 3 x 2 +4 A.)^
1 2 (x^ + 1)(x^ ^ 2)
2.) x^
(^3) 3 x 2 + 2 x 4 B.)^
4(x 1) (x 2) 2
3.) (^42) xx 2 (^) ^44 C.) (^) 2(x (x1)( 2)x+1)
Graph the rational functions given in #4-10. (Their numerators and de- nominators have been completely factored.) Then match each graph with one of the lettered graphs drawn on the next two pages.
4.) 3(x^
(^2) +1) (x 2 +5) 8.)^
(x 4)(x 6) (x 2 +3)(x 2 +4)(x 2 +8)
4(x+1) 2 2(x+2)(x 2) 9.)^
3(x 2 +7) 5(x 2) 2 (x 6)
6.) (x+1)(x^
(^2) +1)(x 2 +8) (x 7) 10.)^
2(x+10) 2 (x+30) 3(x 5)
7.) 7(x + 2) 3 (x 3) 2
11.) Completely factor the numerator and the denominator of the rational function below, and then graph it. Match the graph with one of those on the next two pages.
3 x 3 6 x 2 + x 2 x 2 + 3x + 2
E.) F.)
G.) H.)
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2-
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