Problem 80 - 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. $$y=\frac{x^{4}}{x^{2}-2}$$

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = \pm \sqrt{2}\); x-intercept, y-intercept: \((0,0)\); End behavior polynomial: \(x^2+2\).

Step 1: Identifying the Vertical Asymptotes

To find the vertical asymptotes, set the denominator of the function \(x^2 - 2\) equal to zero and solve for \(x\).\[ x^2 - 2 = 0 \]\[ x^2 = 2 \]\[ x = \pm \sqrt{2} \] Hence, the vertical asymptotes are at \(x = \sqrt{2}\) and \(x = -\sqrt{2}\).

Step 2: Finding the X-Intercepts

The x-intercepts occur where the function's numerator is zero. Since the numerator is \(x^4\), set \(x^4 = 0\). Thus, \(x = 0\) is the x-intercept.

Step 3: Finding the Y-Intercept

The y-intercept occurs where \(x = 0\). Substituting \(x = 0\) into the function yields \(y = \frac{0^4}{0^2 - 2} = 0\). Thus, the y-intercept is at \(y = 0\).

Step 4: Long Division for End Behavior

Perform polynomial long division on \(x^4\) divided by \(x^2 - 2\). Divide \(x^4\) by \(x^2\) to obtain \(x^2\), then multiply \(x^2\) by \(x^2 - 2\). Subtract this result from \(x^4\), continue, and find that the quotient is \(x^2 + 2\). Hence, the polynomial has the same end behavior as the rational function.

Step 5: Description of Graph and Extrema

Graph the rational function and polynomial together. Identify that the polynomials \(x^2 + 2\) follows the same end behavior as the rational function. Local extrema would be determined by setting the derivative equal to zero and testing intervals, but for a rough estimate, these occur where the concavity changes.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes occur where the function approaches infinity as the input (\(x\)) approaches a particular value from either direction. These are places where the function is undefined. For finding the vertical asymptotes of the function \(y=\frac{x^{4}}{x^{2}-2}\), we look at the denominator because division by zero is undefined.
To identify the vertical asymptotes, set the denominator equal to zero:

• \[x^2 - 2 = 0\]
• Solving for \(x\) gives \[x^2 = 2\]
• Thus, \(x = \pm \sqrt{2}\)

This indicates vertical asymptotes at \(x = \sqrt{2}\) and \(x = -\sqrt{2}\). As \(x\) gets closer to these values, the graph of the function shoots up or down indefinitely.

X-Intercepts

X-intercepts occur where a graph crosses the x-axis, meaning the value of the function (\(y\)) is zero. To find x-intercepts in the given function \(y=\frac{x^{4}}{x^{2}-2}\), the numerator must be set to zero because an output of zero indicates that the whole fraction is zero.
Since the numerator here is \(x^4\), setting it to zero gives:

• \[x^4 = 0\]
• This simplifies to \(x = 0\)

So, the x-intercept of this function is just at the origin, \((0, 0)\). This tells you that the graph passes through this point on the x-axis.

Y-Intercepts

Y-intercepts occur where the graph crosses the y-axis, which is equivalent to finding the value of the function when \(x=0\). The point where \(x=0\) will yield the y-value to find this intercept.
For \(y=\frac{x^{4}}{x^{2}-2}\), substitute \(x = 0\) in:

• \[y = \frac{0^4}{0^2 - 2} = 0\]

Thus, the y-intercept is also at \((0, 0)\). This matches the x-intercept, indicating that the graph passes through the origin, and both intercepts are at the same point.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials similar to traditional long division with numbers. It's helpful for understanding the end behavior of rational functions. For the given function \(y=\frac{x^{4}}{x^{2}-2}\), we'll divide the polynomial in the numerator by the polynomial in the denominator.
Here's a brief overview of the process:

• Divide the leading term of the numerator \(x^4\) by the leading term of the denominator \(x^2\). This gives \(x^2\).
• Multiply \(x^2\) by the entire denominator \(x^2 - 2\) and subtract this from the original polynomial \(x^4\).
• Continue the process until no terms of the numerator are left to divide.

The quotient you find is \(x^2 + 2\), which shows you the polynomial with a similar end behavior to the original rational function.

Local Extrema

Local extrema refer to the highest or lowest points in a graph within a specific interval, known as local maxima or minima. In calculus, these are often found using derivatives to determine where the slope changes sign. For our example function, this would involve finding the first derivative and setting it equal to zero to determine potential extrema.
However, the graph can also visually indicate where these points occur by identifying peaks and valleys in the function's curvature.
To better understand local extrema, one would:

• Calculate the first derivative of the function.
• Solve for \(x\) when the derivative equals zero.
• Test values around these \(x\) values to determine if the function switches from increasing to decreasing (or vice versa).

This approach illustrates where the concavity changes, helping identify local extrema without needing graphs for exact precision.