Problem 14 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ F(x)=\frac{1}{x+4} $$

Step-by-Step Solution

Answer

Plot points based on evaluation and sketch considering asymptotes at \( x = -4 \) and \( y = 0 \).

Step 1: Choose Values for x

Select several values of \( x \) around which to evaluate the function. Include negative, zero, and positive values to see behavior around interesting points like where the function might approach vertical asymptotes or cross the x-axis. For this function, we could use \( x = -6, -5, -4, -3, -2, 0, 2 \).

Step 2: Evaluate F(x) for Each x Value

Compute \( F(x) \) for each chosen value. Calculate:- \( F(-6) = \frac{1}{-6+4} = -\frac{1}{2} \)- \( F(-5) = \frac{1}{-5+4} = -1 \)- \( F(-4) \text{ is undefined (division by zero)} \)- \( F(-3) = \frac{1}{-3+4} = 1 \)- \( F(-2) = \frac{1}{-2+4} = \frac{1}{2} \)- \( F(0) = \frac{1}{0+4} = \frac{1}{4} \)- \( F(2) = \frac{1}{2+4} = \frac{1}{6} \)

Step 3: Identify Key Features

Identify features from these calculations: As \( x \to -4 \), \( F(x) \) becomes very large (positive or negative), indicating a vertical asymptote at \( x = -4 \). The function is undefined at \( x = -4 \). As \( x \to \infty \) or \( x \to -\infty \), \( F(x) \to 0 \), indicating a horizontal asymptote at \( y = 0 \).

Step 4: Sketch the Graph

Plot the calculated points on a graph, noting the undefined point at \( x = -4 \) and the vertical and horizontal asymptotes. Connect the points smoothly, considering the asymptotic behavior: the function decreases towards the asymptote as \( x \) approaches \( -4 \) from the left and increases away from the asymptote as \( x \) approaches \( -4 \) from the right.

Key Concepts

Understanding Vertical Asymptotes

A vertical asymptote represents points on a graph where a function's value increases to positive or negative infinity. In the given function, \(F(x) = \frac{1}{x+4}\), the vertical asymptote occurs where the denominator is zero.
This happens when \(x + 4 = 0\), or equivalently, \(x = -4\). At \(x = -4\), the function is undefined, as division by zero is not possible.
When plotting the function, you'll notice that as \(x\) approaches \(-4\) from the left and right, the graph sharply goes up or down without bound. This reveals the characteristic asymptotic behavior where the function values shoot towards infinity.

• The denominator identifies vertical asymptotes.
• At vertical asymptotes, the function isn't defined.
• The behavior near these points significantly impacts graph sketching.

Identifying vertical asymptotes is crucial for understanding where a function graph will consistently fail to touch or cross, giving vital insights for plotting.

Exploring Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \(x\) approaches infinity or negative infinity. For the function \(F(x) = \frac{1}{x+4}\), horizontal asymptotes indicate the value that \(F(x)\) is approaching as \(x\) goes to the far left or right of the graph.

• The horizontal asymptote is calculated by analyzing the behavior of the function at extreme \(x\) values.
• For rational functions like \(F(x)\), compare the degree of the numerator and denominator to determine the asymptote.
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \(y = 0\).

In our function, as \(x\) goes to infinity or negative infinity, \(F(x)\) approaches zero. This means the graph will hover near the line \(y = 0\) but will not actually cross or touch it, signifying the horizontal asymptote. Recognizing horizontal asymptotes aids in understanding the eventual leveling off of the graph's behavior.

Evaluating Functions at Key Points

Evaluating functions simply means substituting specific \(x\) values and calculating to find corresponding function values. This technique builds foundations for plotting functions accurately. In \(F(x) = \frac{1}{x+4}\), evaluating key \(x\) values can help visualize how the graph approximates its shape.

• Select critical points around areas with dramatic changes, like near vertical asymptotes.
• Determine the value of the function at these points to pinpoint where substantial behavior shifts occur.
• For this function, values like \(-6, -5, -3, -2, 0,\) and \(2\) were chosen for demonstrating how function is sketched.

By calculating and plotting these values, one observes \(F(x)\) heading towards infinity and zero, reinforcing understanding of where the graph rises or falls relative to the asymptotes. Evaluating these points is crucial for piecing together the graph's intricate movements.