Problem 5 - 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. $$ y=\frac{1}{x-2}-3 $$

Step-by-Step Solution

Answer

Intercepts are found at \(x = \frac{1}{3} + 2\) and \(y = -3.5\). The vertical asymptote is at \(x = 2\) and the horizontal asymptote is at \(y = -3\). Function does not contain any extrema or inflection points.

Step 1: Identifying the Intercepts

To find the x-intercepts, set y = 0 and solve for x:\(0 = \frac{1}{x-2} - 3\)Multiply through by \(x-2\):\(0 = 1 - 3(x-2)\)Solve for x to give \(x = \frac{1}{3} + 2\)The y-intercept can be found by setting x=0 in the original equation:\(y = \frac{1}{0-2} - 3, y = -3.5\)

Step 2: Identifying the Asymptotes

The denominator cannot be zero so \(x-2\) can't be zero and thus x can't be 2. Therefore, we have a vertical asymptote at x=2. To find the horizontal asymptote, we see what happens to y as x approaches positive and negative infinity:As \(x \rightarrow +\infty, y \rightarrow -3\)As \(x \rightarrow -\infty, y \rightarrow -3\)Thus the horizontal asymptote is y=-3.

Step 3: Relative extrema and Points of inflection

As this is a rational function, it does not have any maximum or minimum points (extrema), neither does it contain points of inflection. This is because the derivative of y is always positive or negative and never equal to zero, thus the slope is always increasing or decreasing.

Step 4: Use a graphing utility to confirm

Plot the function \(y = \frac{1}{x-2} - 3\) using a graphing tool. Validate that the intercepts, asymptotes, and the general function shape match the manual analysis.

Key Concepts

Intercepts

Intercepts are the points where a graph crosses the axes. For the given function, we examine both the x-intercept and y-intercept individually.

To determine the **x-intercept**, set the function equal to zero and solve for x:

• Start with the equation: \[0 = \frac{1}{x-2} - 3\]
• Multiply both sides by \(x-2\): \[0 = 1 - 3(x-2)\]
• Solve for x by simplifying to find: \[x = \frac{1}{3} + 2\]

Thus, the x-intercept of the function is at the coordinate \(\left(\frac{7}{3}, 0\right)\)
To find the **y-intercept**, substitute \(x = 0\) into the function:

• The equation becomes: \[y = \frac{1}{0-2} - 3\]
• Which simplifies to: \[y = -0.5 - 3 = -3.5\]

So, the y-intercept is at the point \((0, -3.5)\). Intercepts are critical to understanding the graph's position relative to the axes.

Asymptotes

Asymptotes are lines that a graph approaches but may never actually reach. This concept gives insight into the end behavior of the function.

**Vertical Asymptote** occurs when the denominator of a rational function equals zero. Here:

• The function denominator is \(x - 2\)
• This means the vertical asymptote is located where \(x = 2\).

At this point, the function is undefined, and the graph approaches the line without crossing it.

**Horizontal Asymptote** describes the behavior of the function as x approaches infinity. For this function:

• As \(x\) approaches either positive or negative infinity, the term \(\frac{1}{x-2}\) becomes negligible.
• Thus, the function approaches \(y = -3\).

The horizontal asymptote is \(y = -3\), indicating where the graph levels off at extreme x-values. Asymptotes help predict the direction and limits of the function's values.

Extrema

Extrema refer to the maximum and minimum values a function can attain either locally or globally. However, for the function \(y = \frac{1}{x-2} - 3\), there are no extrema.

This lack is due to the behavior of the derivative, which tells us about the function's slope:

• The derivative never equals zero or changes sign for rational functions like this.
• This means the graph is either constantly increasing or decreasing without "turning around" to form highs or lows.

In simpler terms, there's no local maximum or minimum points because the slope doesn't change direction.
However, graphically, the vertical asymptote does influence the graph by forming a kind of impassable barrier for the curve. Despite this, it's important to note that it does not create a relative maximum or minimum.

Points of Inflection

Points of inflection occur where the graph of a function changes concavity, or "curves differently." They indicate a shift from concave up to concave down, or vice versa.

For this function, there are no points of inflection. Here's why:

• The second derivative of the function doesn't change sign.
• Without a sign change in the second derivative, there's no switch in concavity.
• Essentially, the graph maintains a consistent form of curvature throughout.

The absence of points of inflection results in a graph that, while dynamic, doesn't have the turning points where its curvature type changes. Instead, it simply approaches its asymptotes and moves through its intercepts steadily.