Asymptote 2 gleiche polynome

In this section we will explore asymptotes of rational functions. In particular, we will look at horizontal, vertical, and oblique asymptotes. Keep in mind that we are studying a rational function of the form,

where P(x) and Q(x) are polynomials. We say that f(x) is in lowest terms if P(x) and Q(x) have no common factors.

1. Horizontal Asymptotes

A. Degree of P(x) < Degree of Q(x)

The rational function f(x) = P(x) / Q(x) in lowest terms has horizontal asymptote
y = 0 if the degree of the numerator, P(x), is less than the degree of denominator, Q(x). In this case, f(x) → 0 as x → ±∞. An example of a function with horizontal asymptote y = 0 is,

B. Degree of P(x) = Degree of Q(x)

The rational function f(x) = P(x) / Q(x) in lowest terms has horizontal asymptote
y = a / b if the degree of the numerator, P(x), is equal to the degree of denominator, Q(x), where a is the leading coefficient of P(x) and b is leading coefficient of Q(x). In this case, f(x) → a / b as x → ±∞. An example of a function with horizontal asymptote y = 1 / 3 is,

C. Degree of P(x) > Degree of Q(x)

The rational function f(x) = P(x) / Q(x) in lowest terms has no horizontal asymptotes if the degree of the numerator, P(x), is greater than the degree of denominator, Q(x).

2. Vertical Asymptotes

The rational function f(x) = P(x) / Q(x) in lowest terms has vertical asymptotes,
x = x1, x = x2, . . . , x = xk, where x1, x2, . . . , xk are roots of Q(x). To find the vertical asymptotes of f(x) be sure that it is in lowest terms by canceling any common factors, and then find the roots of Q(x).

3. Oblique Asymptotes

The rational function f(x) = P(x) / Q(x) in lowest terms has an oblique asymptote if the degree of the numerator, P(x), is exactly one greater than the degree of the denominator, Q(x). You can find oblique asymptotes using polynomial division, where the quotient is the equation of the oblique asymptote. For example, the function,

has oblique asymptote found by polynomial division,

Thus, we found that,

and the equation of the oblique asymptote is the quotient,

y = x + 2.

That is, as |x| gets large, f(x) approaches the line y = x + 2 as depicted in the graph below. Notice that the remainder approaches 0 as |x| → ∞.

Finding Asymptotes

When finding asymptotes, it is important to remember that f(x) = P(x) / Q(x) must be in lowest terms. For example, just because x = a is a simple root of Q(x) does not mean that it corresponds to a vertical asymptote. In particular, if x = a is a simple root of P(x) and Q(x) the graph will have a hole at the point x = a. Since a point has no size, a graph of such a function will not look any different when graphed on a calculator. Therefore, we denote a hole in a graph with an open circle. For example, consider the function,

Notice that we can write Q(x) = x2− 1 as Q(x) = (x − 1)(x + 1). Notice that x = 1 and
x = −1 are not in the domain of f(x) since they are zeros of Q(x). However, if x ≠ 1 we can cancel the factor x − 1 in the numerator and denominator as,

Notice that f(x) andagree everywhere except at the point x = 1 (since x = 1 is not in the domain of f(x), but is in the domain of g(x)). Therefore, the graph of f(x) looks identical to the graph of g(x) except that there is a hole in the graph of f(x) at x = 1 as depicted below.

Consider the rational function where is the degree of the numerator and is the degree of the denominator.

1. If , then the x-axis, , is the horizontal asymptote.

2. If , then the horizontal asymptote is the line .

3. If , then there is no horizontal asymptote (there is an oblique asymptote).

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.

There are three distinct outcomes when checking for horizontal asymptotes:

Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at [latex]y=0[/latex].

Example: [latex]f\left(x\right)=\dfrac{4x+2}{{x}^{2}+4x - 5}[/latex]

In this case the end behavior is [latex]f\left(x\right)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}[/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=\frac{4}{x}[/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[/latex]. Note that this graph crosses the horizontal asymptote.

Horizontal Asymptote [latex]y=0[/latex] when [latex]f\left(x\right)=\dfrac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne{0}\text{ where degree of }p<\text{degree of q}[/latex].

Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.

Example: [latex]f\left(x\right)=\dfrac{3{x}^{2}-2x+1}{x - 1}[/latex]

In this case the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{x}=3x[/latex]. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=3x[/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of [latex]g\left(x\right)=3x[/latex] looks like a diagonal line, and since [latex]f[/latex] will behave similarly to [latex]g[/latex], it will approach a line close to [latex]y=3x[/latex]. This line is a slant asymptote (NOTE: the graph of the function itself is just a “sketch” within the parameters of the asymptotes).

Slant Asymptote when [latex]f\left(x\right)=\dfrac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0[/latex] where degree of [latex]p>\text{ degree of }q\text{ by }1[/latex].

 

To find the equation of the slant asymptote, divide [latex]\dfrac{3{x}^{2}-2x+1}{x - 1}[/latex]. The quotient is [latex]3x+1[/latex], and the remainder is 2. The slant asymptote is the graph of the line [latex]g\left(x\right)=3x+1[/latex].

Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at [latex]y=\frac{{a}_{n}}{{b}_{n}}[/latex], where [latex]{a}_{n}[/latex] and [latex]{b}_{n}[/latex] are the leading coefficients of [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] for [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0[/latex].

Example: [latex]f\left(x\right)=\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[/latex]

In this case the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{{x}^{2}}=3[/latex]. This tells us that as the inputs grow large, this function will behave like the function [latex]g\left(x\right)=3[/latex], which is a horizontal line. As [latex]x\to \pm \infty ,f\left(x\right)\to 3[/latex], resulting in a horizontal asymptote at [latex]y=3[/latex]. Note that this graph crosses the horizontal asymptote.

Horizontal Asymptote when [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0\text{ where degree of }p=\text{degree of }q[/latex].

Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.

It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function

[latex]f\left(x\right)=\dfrac{3{x}^{5}-{x}^{2}}{x+3}[/latex]

with end behavior

[latex]f\left(x\right)\approx \dfrac{3{x}^{5}}{x}=3{x}^{4}[/latex],

the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.

As [latex]x\to \pm \infty , f\left(x\right)\to \infty [/latex]

 

A General Note: Horizontal Asymptotes of Rational Functions

The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.

• Degree of numerator is less than degree of denominator: horizontal asymptote at
• [latex]y=0[/latex]
• Degree of numerator is greater than degree of denominator by one: no horizontal asymptote; slant asymptote.
• Degree of numerator is equal to degree of denominator: horizontal asymptote at ratio of leading coefficients.

Example: Identifying Horizontal and Slant Asymptotes

For the functions below, identify the horizontal or slant asymptote.

1. [latex]g\left(x\right)=\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[/latex]
2. [latex]h\left(x\right)=\dfrac{{x}^{2}-4x+1}{x+2}[/latex]
3. [latex]k\left(x\right)=\dfrac{{x}^{2}+4x}{{x}^{3}-8}[/latex]

Show Solution

For these solutions, we will use [latex]f\left(x\right)=\dfrac{p\left(x\right)}{q\left(x\right)}, q\left(x\right)\ne 0[/latex].

1. [latex]g\left(x\right)=\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[/latex]: The degree of [latex]p[/latex] and the degree of [latex]q[/latex] are both equal to 3, so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at [latex]y=\frac{6}{2}[/latex] or [latex]y=3[/latex].
2. [latex]h\left(x\right)=\dfrac{{x}^{2}-4x+1}{x+2}[/latex]: The degree of [latex]p=2[/latex] and degree of [latex]q=1[/latex]. Since [latex]p>q[/latex] by 1, there is a slant asymptote found at [latex]\dfrac{{x}^{2}-4x+1}{x+2}[/latex].
   
   The quotient is [latex]x - 6[/latex] and the remainder is 13. There is a slant asymptote at [latex]y=x - 6[/latex].
3. [latex]k\left(x\right)=\dfrac{{x}^{2}+4x}{{x}^{3}-8}[/latex]: The degree of [latex]p=2\text{ }<[/latex] degree of [latex]q=3[/latex], so there is a horizontal asymptote [latex]y=0[/latex].

Try it

Watch this video to see more worked examples of determining which kind of horizontal asymptote a rational function will have.

Example: Identifying Horizontal Asymptotes

In the sugar concentration problem earlier, we created the equation [latex]C\left(t\right)=\dfrac{5+t}{100+10t}[/latex].

Find the horizontal asymptote and interpret it in context of the problem.

Show Solution

Both the numerator and denominator are linear (degree 1). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is [latex]t[/latex], with coefficient 1. In the denominator, the leading term is [latex]10t[/latex], with coefficient 10. The horizontal asymptote will be at the ratio of these values:

[latex]t\to \infty , C\left(t\right)\to \frac{1}{10}[/latex]

This function will have a horizontal asymptote at [latex]y=\frac{1}{10}[/latex].

This tells us that as the values of [latex]t[/latex] increase, the values of [latex]C[/latex] will approach [latex]\frac{1}{10}[/latex]. In context, this means that, as more time goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or [latex]\frac{1}{10}[/latex] pounds per gallon.

Example: Identifying Horizontal and Vertical Asymptotes

Find the horizontal and vertical asymptotes of the function

[latex]f\left(x\right)=\dfrac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}[/latex]

Show Solution

First, note that this function has no common factors, so there are no potential removable discontinuities.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\text{and }5[/latex], indicating vertical asymptotes at these values.

The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\to \pm \infty , f\left(x\right)\to 0[/latex]. This function will have a horizontal asymptote at [latex]y=0[/latex].

Try It

Find the vertical and horizontal asymptotes of the function:

[latex]f\left(x\right)=\dfrac{\left(2x - 1\right)\left(2x+1\right)}{\left(x - 2\right)\left(x+3\right)}[/latex]

Show Solution

Vertical asymptotes at [latex]x=2[/latex] and [latex]x=-3[/latex]; horizontal asymptote at [latex]y=4[/latex].



A General Note: Intercepts of Rational Functions

A rational function will have a y-intercept when the input is zero, if the function is defined at zero. A rational function will not have a [latex]y[/latex]-intercept if the function is not defined at zero.

Likewise, a rational function will have [latex]x[/latex]-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, [latex]x[/latex]-intercepts can only occur when the numerator of the rational function is equal to zero.

Example: Finding the Intercepts of a Rational Function

Find the intercepts of [latex]f\left(x\right)=\dfrac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}[/latex].

Show Solution

We can find the [latex]y[/latex]-intercept by evaluating the function at zero

[latex]\begin{align}f\left(0\right)&=\dfrac{\left(0 - 2\right)\left(0+3\right)}{\left(0 - 1\right)\left(0+2\right)\left(0 - 5\right)} \\[1mm] &=\frac{-6}{10} \\[1mm] &=-\frac{3}{5} \\[1mm] &=-0.6 \end{align}[/latex]

The [latex]x[/latex]-intercepts will occur when the function is equal to zero. A rational function is equal to 0 when the numerator is 0, as long as the denominator is not also 0.

[latex]\begin{align} 0&=\frac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)} \\[1mm] 0&=\left(x - 2\right)\left(x+3\right)\end{align}[/latex]

[latex]x=2, -3[/latex]

The [latex]y[/latex]-intercept is [latex]\left(0,-0.6\right)[/latex], the [latex]x[/latex]-intercepts are [latex]\left(2,0\right)[/latex] and [latex]\left(-3,0\right)[/latex].

Try It

Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the [latex]x[/latex]– and [latex]y[/latex]-intercepts and the horizontal and vertical asymptotes.

Show Solution

For the transformed reciprocal squared function, we find the rational form.

[latex]\begin{align}f\left(x\right)&=\dfrac{1}{{\left(x - 3\right)}^{2}}-4 \\[1mm] &=\dfrac{1 - 4{\left(x - 3\right)}^{2}}{{\left(x - 3\right)}^{2}} \\[1mm] &=\dfrac{1 - 4\left({x}^{2}-6x+9\right)}{\left(x - 3\right)\left(x - 3\right)} \\[1mm] &=\frac{-4{x}^{2}+24x - 35}{{x}^{2}-6x+9}\end{align}[/latex]

Because the numerator is the same degree as the denominator we know that as [latex]x\to \pm \infty , f\left(x\right)\to -4; \text{so } y=-4[/latex] is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is [latex]x=3[/latex], because as [latex]x\to 3,f\left(x\right)\to \infty[/latex]. We then set the numerator equal to 0 and find the [latex]x[/latex]-intercepts are at [latex]\left(2.5,0\right)[/latex] and [latex]\left(3.5,0\right)[/latex]. Finally, we evaluate the function at 0 and find the [latex]y[/latex]-intercept to be at [latex]\left(0,\frac{-35}{9}\right)[/latex].

Watch the following video to see more worked examples of finding asymptotes, intercepts and holes of rational functions.