Problem 73 - 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. \(r(x)=\frac{x^{4}-3 x^{3}+6}{x-3}\)

Step-by-Step Solution

Answer

Vertical asymptote at \(x=3\), \(y\)-intercept at \((0,-2)\), and end behavior of \(x^{3}\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero, provided that the numerator does not also become zero at these points. For the function \(r(x)=\frac{x^{4}-3x^{3}+6}{x-3}\), the denominator \(x-3=0\) creates a vertical asymptote at \(x=3\). Check the numerator at \(x=3\): \(x^{4}-3x^{3}+6 = 3^4 - 3 \times 3^3 + 6 = 81 - 81 + 6 = 6 eq 0\). Therefore, there is a vertical asymptote at \(x=3\).

Step 2: Find Intercepts

To find the \(x\)-intercepts, set the numerator equal to zero: \(x^{4} - 3x^{3} + 6 = 0\). Solving this quartic equation requires numerical methods or graphing software, as it does not factor easily. For the \(y\)-intercept, set \(x=0\) in the function: \(r(0) = \frac{0^{4} - 3 \times 0^{3} + 6}{0-3} = -2\). Thus, the \(y\)-intercept is \((0, -2)\).

Step 3: Calculate Local Extrema

Differentiate \(r(x)\) to find its critical points. The derivative of \(r(x)=\frac{x^{4}-3x^{3}+6}{x-3}\) using the quotient rule is \(r'(x) = \frac{(4x^{3} - 9x^{2})(x-3) - (x^{4} - 3x^{3} + 6)}{(x-3)^{2}}\). Solve \(r'(x) = 0\) using numerical methods to find critical points, as this does not simplify easily. Evaluate these critical points to determine local maxima or minima.

Step 4: Perform Long Division

Perform polynomial long division of \(x^{4} - 3x^{3} + 6\) by \(x-3\). Divide the leading term \(x^{4}\) by \(x\) to get \(x^{3}\), multiply \(x^{3}\) by \(x-3\), subtract from the original polynomial, and repeat this process for the remainder terms until the degree of the remainder is less than the degree of the divisor. This yields \(r(x) = x^{3}\) with a remainder of \(6\).

Step 5: Graph Both Functions

Graph the original rational function \(r(x) = \frac{x^{4} - 3x^{3} + 6}{x-3}\) and the polynomial \(x^{3}\) on the same set of axes. Ensure the viewing window is large enough to observe the behavior as \(x\) approaches positive and negative infinity. Both graphs will approach the same values, confirming that the end behavior is the same. The graph will show that as \(x\to \pm\infty\), both functions approach \(x^{3}\).

Key Concepts

Vertical Asymptotes

When trying to graph a rational function, one of the first steps is to determine the vertical asymptotes. A vertical asymptote occurs when the denominator of the function equals zero, causing the function to be undefined at a particular point, provided the numerator doesn't also zero out at the same place. For our rational function, \( r(x)=\frac{x^{4}-3x^{3}+6}{x-3} \), we set the denominator equal to zero: \( x-3=0 \), which gives us \( x=3 \). At this point, if we substitute back into the numerator, \( x^{4} - 3x^{3} + 6 \) yields \( 6 \) when \( x=3 \), confirming that the numerator doesn't equal zero. Thus, there is a vertical asymptote at \( x=3 \). This tells us that the function's graph will have a break at \( x=3 \) and will approach infinity either positively or negatively as \( x \) approaches this value from the left and right.

X-Intercepts

Finding the \( x \)-intercepts of a rational function involves setting the numerator equal to zero and solving for \( x \). For the function \( r(x) = \frac{x^{4} - 3x^{3} + 6}{x-3} \), we solve \( x^{4} - 3x^{3} + 6 = 0 \). This is a quartic equation, which can be quite complex to solve algebraically. In most cases, numerical methods or graphing calculators/software can assist in finding approximate solutions. The solutions to this equation represent the \( x \)-values where the graph crosses or touches the \( x \)-axis. Since the exact intercepts are not easily factorable by hand, approximation tools are often necessary for practical applications.

Y-Intercepts

The \( y \)-intercept of a function is found by evaluating the function when \( x=0 \). For our rational function \( r(x)=\frac{x^{4}-3x^{3}+6}{x-3} \), substituting \( x=0 \) gives us: \( r(0) = \frac{0^{4} - 3(0)^{3} + 6}{0-3} = \frac{6}{-3} = -2 \). Thus, the \( y \)-intercept is the point \( (0, -2) \). This tells us that the graph of the function will cross the \( y \)-axis at \( -2 \), providing a useful starting point for sketching or graphing the function manually or using software.

Local Extrema

Local extrema of a function, such as local maxima or minima, occur at critical points where the derivative equals zero or is undefined. To find them in our function, we need to differentiate \( r(x) \) using the quotient rule: \( r'(x) = \frac{(4x^{3} - 9x^{2})(x-3) - (x^{4} - 3x^{3} + 6)}{(x-3)^{2}} \). Set this derivative equal to zero to find critical points. Due to the complexity of simplifying this derivative analytically, numerical methods or graphing software are typically used to pinpoint these critical points accurately. Evaluating these points will help us determine whether they correspond to local maxima, minima, or possibly saddle points.

Polynomial Long Division

Using polynomial long division helps us find a simpler polynomial function that mirrors the end behavior of our rational function. The process involves dividing the numerator, \( x^{4} - 3x^{3} + 6 \), by the denominator, \( x-3 \). Begin by dividing the leading term \( x^{4} \) by \( x \), resulting in \( x^{3} \). Multiply \( x^{3} \) by \( x-3 \), subtract from the original polynomial, and continue this process for each remainder. For this function, you end up with \( r(x) = x^{3} \) with a remainder of \( 6 \). Thus, as \( x \) approaches infinity in either direction, both \( r(x) \) and the polynomial \( x^{3} \) will follow the same trajectory, confirming identical end behavior.