Problem 79 - Essential Calculus

Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x-2}{x^{2}-4 x+3} $$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x-2}{x^2 - 4x + 3}\) has vertical asymptotes at \(x = 1\) and \(x = 3\), and a horizontal asymptote at \(y = 0\). The function does not have any extrema, as confirmed by the derivative analysis and the graph.

Step 1: Analyzing Asymptotes

Analyze the function for vertical and horizontal asymptotes. To find the vertical asymptote(s), set the denominator of the function equal to zero and solve for \(x\). Therefore, \(x^2 - 4x + 3 = 0\). This can be factored into \((x-3)(x-1) = 0\), yielding the roots \(x = 1\) and \(x = 3\). These are the vertical asymptotes. For the horizontal asymptote, examine the degrees of the numerator and denominator. If the degree of the numerator is less than that of the denominator, then the \(y\)-axis is a horizontal asymptote. Therefore, \(y = 0\) is the horizontal asymptote.

Step 2: Finding Extrema

Find the critical numbers of the function by taking the derivative and setting it equal to zero. The derivative of \(f(x) = \frac{x-2}{x^2 - 4x + 3}\) is \(f'(x) = \frac{-x^2 + 8x - 7}{(x^2 - 4x +3)^2}\). Setting this equation to zero we get no real solutions, suggesting there are no extrema.

Step 3: Graph and Label

Graph the function and plot the asymptotes and label them. The plot would clearly show the absence of any local maxima or minima, and it would approach the line \(y = 0\) as \(x\) approaches plus or minus infinity. It would also show that the function is undefined at \(x = 1\) and \(x = 3\). Label all these details in your graph.

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They help to determine the behavior of a function at its extremes.

• Vertical asymptotes occur where the function is undefined, typically where the denominator equals zero. In our example, these occur at the roots of the denominator equation: \((x^2 - 4x + 3 = 0)\), which factors to \((x-3)(x-1) = 0\) giving us the x-values \(x = 1\) and \(x = 3\).


• Horizontal asymptotes describe how the function behaves as \(x\) approaches infinity. If the numerator has a lower degree than the denominator, as in this function, the horizontal asymptote is the x-axis or \(y = 0\).

Understanding and labeling asymptotes is crucial in the graphical analysis as it highlights where a function may be undefined or tends to level off.
They form boundaries that the graph approaches closely but never actually meets.

Extrema

Extrema refer to the maximum and minimum values of a function. They are critical in identifying where a function reaches its highest or lowest points. To find extrema, we typically look for critical points and evaluate them.
In this particular example, through derivative analysis, we've determined that no real solutions exist where the derivative is zero.

• This means the function has no local maxima or minima.

Graphical analysis shows how the function behaves, and lack of extrema indicates a continuous increase or decrease without specific peaks or valleys.

Critical Points

Critical points are where the derivative of the function equals zero or is undefined. They are significant because they may represent locations of possible extrema.
For the function \(f(x) = \frac{x-2}{x^2 - 4x + 3}\), critical points are checked by setting the derivative equal to zero.

• The derivative is \(f'(x) = \frac{-x^2 + 8x - 7}{(x^2 - 4x + 3)^2}\).
• In our solution, no real solutions are found when setting \(f'(x) = 0\), which means there are no critical points for extrema.

Critical points are key in understanding the potential shifts in the concavity and direction of the graph. Despite the lack of critical points in our specific function, they are useful in broader contexts to locate key features of other graphs.

Derivative

The derivative of a function gives us the rate at which the function changes. In graphical analysis, it is useful for understanding the slope and behavior of function graphs.
For the function \(f(x) = \frac{x-2}{x^2 - 4x + 3}\), the derivative \(f'(x) = \frac{-x^2 + 8x - 7}{(x^2 - 4x + 3)^2}\) helps reveal important characteristics:

• By taking the derivative, we ascertain the slope of the tangent line at any point \(x\).
• Setting \(f'(x) = 0\) typically highlights potential extrema, although in this case, no real solutions exist.

Studying derivatives equips us with insights into the growth patterns and intervals of increase or decrease of functions on a graph.

Computer Algebra System

A computer algebra system (CAS) is a software tool that aids in symbolic mathematical computations. It is particularly useful for complex algebraic problems that can be cumbersome to solve manually.
With features that automatically compute derivatives, integrals, and factorizations, CAS efficiently helps in graphing functions and analyzing their properties.

• In the given exercise, using a CAS simplifies solving for asymptotes and determining the derivative.
• A CAS quickly processes symbolic expressions to aid in graphical analysis.

These systems are incredibly useful in higher mathematics for providing accurate and swift solutions to potentially error-prone manual calculations.