Problem 13 - Calculus of a Single Variable

Question

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=\frac{x^{2}-6 x+12}{x-4} $$

Step-by-Step Solution

Answer

The y-intercept is at (0, -3), there is a vertical asymptote at \(x=4\) and a horizontal asymptote at \(y=x\). The function does not have x-intercepts, relative extrema, or points of inflection. The function decreases on \((-∞, 4)\) and increases on \((4, ∞)\), approaching the asymptotes.

Step 1: Finding the intercepts

To find the y-intercept, let \(x = 0\) in the equation and resolve for \(y\): \(y=\frac{(0)^{2}-6(0)+12}{0-4}=-3\). Therefore, the y-intercept at is (0,-3). To find the x-intercept, set \(y = 0\) and solve for \(x\), however the numerator must be zero which results in \(x^{2}-6x+12=0\). The are no real roots for this equation, so no x-intercept exists.

Step 2: Locating the asymptotes

The function \(y=\frac{x^{2}-6x+12}{x-4}\) is undefined for \(x=4\), so there's a vertical asymptote at \(x=4\). To find the horizontal asymptote, examine the degrees of the polynomial in the numerator and denominator. They're equal, so the horizontal asymptote is the ratio of the coefficients of the highest degree terms, hence, \(y= x\).

Step 3: Determining the extrema and inflection points

To determine extrema, you must first find the derivative of the function: \(y' =\frac{(2x-6)*(x-4)-(x^{2}-6x+12)}{(x-4)^{2}}\), and then set it to zero to find the critical points. Unfortunately, the derivative never equals zero. Without any inflection points or extrema, the only change in concavity will occur at vertical asymptotes.

Step 4: Sketching the graph

By incorporating the y-intercept from step 1, the vertical asymptote from step 2 and the information about extrema from step 3, sketch the graph. Since no extrema are found, the function is monotonic, decreasing on \((-∞, 4)\) and increasing on \((4, ∞)\) while approaching the asymptotes.

Step 5: Verifying with graphing utility

To confirm these results, plot the function on a graphing utility and verify the intercepts, asymptotes, and general shape.

Key Concepts

Finding Intercepts

When sketching the graph of a rational function, the intercepts provide essential starting points. To find the y-intercept, we set the x-value to zero and solve for y. For the given function, we calculate the y-intercept by substituting 0 for x, yielding \( y=\frac{(0)^{2}-6(0)+12}{0-4}=-3 \). This point, (0, -3), will be a spot where the graph crosses the y-axis.

Finding x-intercepts involves setting the y-value to zero which means the numerator must be zero, but solving \( x^{2}-6x+12=0 \) reveals no real solutions. Consequently, our graph does not cross the x-axis at any point. Understanding intercepts allows us to anchor our graph accurately on the coordinate system.

Asymptote Analysis

Asymptotes are invisible lines that the graph of a function approaches but never touches. They are a critical aspect in shaping our graph. For our function, \( y=\frac{x^{2}-6x+12}{x-4} \), a vertical asymptote exists at \( x=4 \) because the function is undefined there. The graph will get infinitely close to this line but never cross it.

Horizontal asymptotes are found by comparing the degrees of the polynomials in the numerator and denominator. Since they are equal here, the horizontal asymptote will be the ratio of the leading coefficients, which gives us a horizontal asymptote at \( y=x \). Our graph will level out to this line as \( x \) moves towards positive or negative infinity.

Derivative for Extrema and Inflection Points

To identify the relative extrema and inflection points of a function, we use its derivative. For this example, the derivative is found to be \( y' =\frac{(2x-6)*(x-4)-(x^{2}-6x+12)}{(x-4)^{2}} \). Normally, we would set this derivative to zero to solve for critical points, which represent potential maxima or minima. However, in this case, the derivative does not equal zero for any value of x, meaning there are no extrema.

Furthermore, points of inflection occur where concavity changes, which requires a second derivative test. Unfortunately, the function does not have points of inflection either. Thus, there will be no relative minimums or maximums, and the function will be monotonic within the partitions created by the vertical asymptote.

Using Graphing Utilities

Graphing utilities are incredibly useful for verifying our findings when analyzing rational functions. After manually sketching the graph based on intercepts and asymptotes, we can use a tool like a graphing calculator or software to plot \( y=\frac{x^{2}-6x+12}{x-4} \). By comparing our graph to what's displayed on the utility, we confirm the y-intercept, vertical and horizontal asymptotes, and the overall shape of the curve.

These utilities can also highlight nuances that might be missed by hand, such as subtle changes in curvature or the precise nature of asymptotic behavior. They serve as an excellent means to conclude our graph analysis with confidence in our understanding and representation of the function's behavior.