Problem 43 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=\frac{-2}{x+2}+2$$

Step-by-Step Solution

Answer

Graph the rational function, noting asymptotes at \( x = -2 \) and \( y = 2 \).

Step 1: Identify the function type

The given function is \( f(x) = \frac{-2}{x+2} + 2 \). This is a rational function, where the main component \( \frac{-2}{x+2} \) is a hyperbola, and it is shifted vertically and horizontally.

Step 2: Determine the vertical asymptote

Rational functions have vertical asymptotes where the denominator is zero. For \( \frac{-2}{x+2} \), the vertical asymptote is at \( x + 2 = 0 \), which simplifies to \( x = -2 \).

Step 3: Determine the horizontal asymptote

As \( x \to \pm \infty \), the function \( \frac{-2}{x+2} \to 0 \). Hence, the horizontal asymptote of the function \( f(x) = \frac{-2}{x+2} + 2 \) is determined by the constant term \( +2 \). Therefore, the horizontal asymptote is \( y = 2 \).

Step 4: Evaluate the function at key points

Evaluate the function at points of interest to aid in plotting. For example:- When \( x = 0 \), \( f(0) = \frac{-2}{0+2} + 2 = 1 \)- When \( x = -3 \), \( f(-3) = \frac{-2}{-3+2} + 2 = 4 \).

Step 5: Plot the asymptotes and points

Draw the vertical asymptote as a dashed line at \( x = -2 \) and the horizontal asymptote as a dashed line at \( y = 2 \). Mark the points determined in Step 4, such as \((0, 1)\) and \((-3, 4)\).

Step 6: Sketch the graph

Using the asymptotes and the plotted points, sketch the hyperbola. The graph will approach the asymptotes but never touch them. The portion of the graph left of \( x = -2 \) descends above \( y = 2 \), and the portion right of \( x = -2 \) descends below \( y = 2 \).

Key Concepts

Vertical Asymptote

A vertical asymptote for a rational function occurs at points where the function's denominator is zero. These are the values of \( x \) that would make the function undefined. In simpler terms, it's where the graph of the function shoots up to infinity or down to negative infinity. For our function \( f(x) = \frac{-2}{x+2} + 2 \), the denominator is \( x+2 \).

• Set \( x+2 = 0 \) to find the vertical asymptote.
• This gives \( x = -2 \).

This means the graph has a vertical asymptote at \( x = -2 \). When graphing, this is typically shown as a dashed line because the function approaches this line but never crosses it.

Horizontal Asymptote

Horizontal asymptotes occur in rational functions when the function tends towards a constant value as \( x \) approaches infinity or negative infinity. For the same function, \( f(x) = \frac{-2}{x+2} + 2 \), the horizontal asymptote is determined by examining how the function behaves as \( x \to \pm \infty \).

• The main component, \( \frac{-2}{x+2} \), tends to 0 as \( x \) becomes very large or very small.
• The constant \(+2\) remains unaffected by changes in \( x \).

So, the entire function tends towards \( y = 2 \) as \( x \to \pm \infty \). Hence, \( y = 2 \) is the horizontal asymptote, indicating that the graph will run parallel to this line without touching it.

Plotting Points

Plotting specific points helps in creating a more accurate graph of the rational function. You can choose some values for \( x \) and substitute them into the function to find corresponding \( y \) values. This helps in mapping out the shape and position of the graph around the asymptotes.

• For \( x = 0 \): Substitute \( x \) in \( f(x) = \frac{-2}{x+2} + 2 \), giving \( f(0) = 1 \). So the point is \((0, 1)\).
• For \( x = -3 \): Substitute \( x \) in \( f(x) = \frac{-2}{x+2} + 2 \), yielding \( f(-3) = 4 \). So the point is \((-3, 4)\).

By plotting these points, we can see how the graph behaves around the vertical asymptote and how it approaches the horizontal asymptote.

Asymptotic Behavior

The term asymptotic behavior describes how the graph of the function behaves as it approaches the asymptotes. This behavior helps us to predict the path of the graph as \( x \) moves towards the vertical and horizontal asymptotes.

• Near the vertical asymptote \( x = -2 \), the graph will curve steeply upwards or downwards depending on the side of \( x = -2 \) we are viewing it from.
• As \( x \to \infty \) or \( x \to -\infty \), the graph approaches the horizontal asymptote \( y = 2 \) but never actually reaches it.

Understanding this behavior is crucial to drawing an accurate representation of the function, as it shows the transition of the curve, indicating that the line continuously approaches but never quite touches the asymptotes.