Problem 82 - Precalculus Mathematics for Calculus

Question

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = \pm 1 \), y-intercept at \( (0, -4) \). End behavior matches polynomial from division.

Step 1: Identify Vertical Asymptotes

To find the vertical asymptotes of the rational function \( r(x) = \frac{4 + x^2 - x^4}{x^2 - 1} \), we need to determine where the denominator is zero. Set \( x^2 - 1 = 0 \), solved by \( x = \pm 1 \). Therefore, the vertical asymptotes are at \( x = 1 \) and \( x = -1 \).

Step 2: Locate x-Intercepts

x-intercepts occur where the numerator is zero. Set \( 4 + x^2 - x^4 = 0 \). Rewrite it as \( -x^4 + x^2 + 4 = 0 \). Solving this equation can provide values for \( x \); here it shows potential intercepts.

Step 3: Determine y-Intercept

The y-intercept is found by evaluating \( r(x) \) at \( x = 0 \). Plug \( x = 0 \) into the function: \( r(0) = \frac{4 + 0^2 - 0^4}{0^2 - 1} = \frac{4}{-1} = -4 \). Thus, the y-intercept is \( (0, -4) \).

Step 4: Identify Local Extrema

To find local extrema, find the first derivative of \( r(x) \) and determine where it equals zero or is undefined, then use the first or second derivative test. Extrema can be determined by setting the derivative to zero and solving for \( x \), but typically involves calculus tools beyond algebratic solutions.

Step 5: Determine End Behavior via Long Division

To find a polynomial with the same end behavior as the rational function, use long division to divide \( -x^4 + x^2 + 4 \) by \( x^2 - 1 \). The result gives the polynomial describing the end behavior.

Step 6: Graph the Functions

Using software or graph paper, graph the function \( r(x) \) and the polynomial found in Step 5 to visually confirm the matching end behavior in a sufficiently large viewing angle.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes occur where the denominator equals zero, provided the numerator is not zero simultaneously. For the function \[ r(x) = \frac{4 + x^2 - x^4}{x^2 - 1} \]you'll first solve for where the denominator equals zero. So you set:- \( x^2 - 1 = 0 \)- Which results in \( x = \pm 1 \).These solutions indicate that there are vertical asymptotes at \( x = 1 \) and \( x = -1 \). On the graph, as you approach these values from either side, the function's value tends to infinity or negative infinity. Make sure to note that a graph will never cross these lines, they act like invisible barriers.

X-Intercepts

The x-intercepts of a rational function occur where the numerator is equal to zero. For the given rational function:\[ 4 + x^2 - x^4 = 0 \]You need to solve for \(x\) to find where the graph crosses the x-axis. This is often more complex, as it might involve solving a polynomial equation. The x-intercepts are the real solutions to this equation. In some cases, factoring or using methods like the quadratic formula might help. Once you have the x-values, these are the points where \( y = 0 \) on the graph.

Y-Intercepts

To find the y-intercept, evaluate the rational function when \( x = 0 \). This is because the y-intercept occurs where the graph crosses the y-axis, so you need the output of the function at \( x = 0 \). For our function:\[ r(0) = \frac{4 + 0^2 - 0^4}{0^2 - 1} = -4 \]Thus, the y-intercept is at the point \( (0, -4) \). This point tells you where the graph will cross the y-axis, and can be an essential aspect when sketching the function.

Local Extrema

Local extrema in rational functions are points where the function reaches a local minimum or maximum. To find these points accurately, calculus tools like derivatives are usually employed. The first derivative test helps identify these extrema by finding where the derivative is zero or undefined, indicating potential peaks or troughs. It's a process of examining where the slope of the tangent to the curve is zero.- Calculate the first derivative of the function.- Set the derivative equal to zero and solve for \( x \) to find potential extrema.- Use either the first or second derivative test to categorize these points as maxima, minima, or saddle points.These steps might require algebraic manipulation and knowledge of calculus.

End Behavior

The end behavior of a rational function describes how the function behaves as \( x \) approaches infinity or negative infinity. Often, this can be understood by examining the leading terms of the numerator and denominator. However, a more accurate approach might be using polynomial long division. - Perform long division of the numerator by the denominator in the rational function.- The quotient obtained gives a polynomial that approximates the rational function's end behavior.For example, if you divide \[ -x^4 + x^2 + 4 \]by \[ x^2 - 1 \],the resulting polynomial describes the function as \( x \to \pm \infty \). This step helps determine if the rational function approaches a line or a constant as \( x \to \infty \).

Long Division

Long division in mathematics is similar to regular division you might be familiar with, but involves polynomials instead of numbers. This is crucial when you're trying to simplify complex rational expressions or analyze their end behavior.- To perform polynomial long division, divide the highest degree term of the numerator by the highest degree term of the denominator.- Write down the result, then multiply the entire divisor by this result and subtract from the original numerator.- Repeat this process with the remainder until you've divided fully or achieved a degree lower than the divisor.In the case of the function at hand, dividing \(-x^4 + x^2 + 4\) by \( x^2 - 1 \) helps in getting a clearer picture of the end behavior by simplifying the rational function to an easier-to-interpret polynomial approximation for large values of \( x \). This technique is particularly helpful in sketching graphs of rational functions.