Problem 72 - Algebra and Trigonometry

Question

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct 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 at \( x = \pm \sqrt{2} \); intercepts at \((0,0)\); end behavior \( y \approx x^2 + 2 \).

Step 1: Identify Vertical Asymptotes

The vertical asymptotes occur where the denominator is zero and the numerator is not zero. For the function \( y = \frac{x^4}{x^2 - 2} \), set the denominator equal to zero and solve for \( x \): \( x^2 - 2 = 0 \). Solving this, we get \( x = \pm \sqrt{2} \). These are your vertical asymptotes.

Step 2: Find the x-intercepts

The \( x \)-intercepts occur where the numerator is zero. For the function \( y = \frac{x^4}{x^2 - 2} \), set the numerator equal to zero: \( x^4 = 0 \), which gives \( x = 0 \). The \( x \)-intercept is at (0,0).

Step 3: Find the y-intercepts

The \( y \)-intercepts are found by setting \( x = 0 \) and calculating \( y \). For the function \( y = \frac{x^4}{x^2 - 2} \), substituting \( x = 0 \) gives \( y = \frac{0}{-2} = 0 \). Therefore, the \( y \)-intercept is also at (0,0).

Step 4: Determine Local Extrema

To find local extrema, calculate the derivative of \( y = \frac{x^4}{x^2 - 2} \) and find critical points by setting \( y' = 0 \). The derivative is calculated using the quotient rule: \( y' = \frac{d/dx(x^4) \cdot (x^2-2) - x^4 \cdot d/dx(x^2-2)}{(x^2-2)^2} \). Simplify and solve \( y' = 0 \) to find critical points, then use the second derivative test or analyze the intervals for any local maxima or minima.

Step 5: Use Long Division for End Behavior

Perform polynomial long division with \( x^4 \) divided by \( x^2 - 2 \) to determine the polynomial that represents the end behavior of the rational function. The division gives \( x^2 + 2 \), indicating that as \( x \to \pm \infty \), \( y \approx x^2 + 2 \).

Step 6: Graphing and Verification

Graph the function \( y = \frac{x^4}{x^2-2} \) and the polynomial \( y = x^2 + 2 \) on a large viewing window to verify their end behavior. Observe that as \( |x| \to \infty \), the function behaves like the polynomial, confirming the match in end behavior.

Key Concepts

Vertical Asymptotes

Vertical asymptotes of a rational function occur where the denominator equals zero, making the function undefined at that point. For the given function, \( y = \frac{x^4}{x^2 - 2} \), we find the vertical asymptotes by setting the denominator \( x^2 - 2 \) to zero.

• Set the denominator equal to zero: \( x^2 - 2 = 0 \).
• Solve for \( x \): \( x = \pm \sqrt{2} \).

These two values, \( x = \sqrt{2} \) and \( x = -\sqrt{2} \), are the vertical asymptotes of the function. At these points, the graph will approach these lines but will not cross them, demonstrating a break in the function's continuity.

X-Intercepts

The \( x \)-intercepts of a rational function are points where the graph crosses or touches the x-axis. This occurs when the numerator is zero, as this will make the entire fraction equal to zero. In the function \( y = \frac{x^4}{x^2 - 2} \), we set the numerator \( x^4 \) to zero and solve for \( x \).

• Set numerator equal to zero: \( x^4 = 0 \).
• Since \( x^4 \) equals zero when \( x = 0 \), this means the function crosses the x-axis at (0,0).

Thus, the only \( x \)-intercept for this function is at the origin, which is a point where the graph both touches and crosses the x-axis.

Y-Intercepts

Y-intercepts are points where the graph crosses the y-axis, and these are found by setting \( x = 0 \) in the equation. In the function \( y = \frac{x^4}{x^2 - 2} \), substituting \( x = 0 \) allows us to calculate the y-value:

• Substitute \( x = 0 \) into the function: \( y = \frac{0^4}{0^2 - 2} = 0 \).

The result is \( y = 0 \), indicating that the y-intercept is also at (0,0). It's not uncommon for the \( x \)-intercept and \( y \)-intercept to be the same for a rational function, particularly when the graph passes through the origin.

End Behavior

The end behavior of a rational function describes how the graph behaves as \( x \) approaches positive or negative infinity. For our function \( y = \frac{x^4}{x^2 - 2} \), to find the end behavior, we use polynomial long division on the fraction: divide \( x^4 \) by \( x^2 - 2 \).

• Through long division, we find the quotient \( x^2 + 2 \).

As \( x \to \pm \infty \), the leading behavior of the function approaches that of the polynomial \( y = x^2 + 2 \). This result indicates that the graph of the rational function will follow the same \( U \)-shape pattern as the parabola \( y = x^2 + 2 \) at extreme values of \( x \), confirming that these functions have matching end behavior.