Problem 50 - College Algebra

Question

Use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+2}-2$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\frac{1}{x+2}-2\) is the graph of the function \(f(x)=\frac{1}{x}\) shifted 2 units to the left and 2 units down, with asymptotes now at \(x=-2\) and \(y=-2\).

Step 1: Initial Function Understanding

The given function is \(g(x)=\frac{1}{x+2}-2\). The baseline for this function would be \(f(x)=\frac{1}{x}\) (a hyperbola) from the request.

Step 2: Understanding Transformation

Next, it's important to understand how transformations work. Adding a constant inside the function \(f(x+a)\) will shift the graph to the left by \(a\) units, while adding a constant outside the function \(f(x) + a\) will shift the graph upwards by \(a\) units.

Step 3: Function Transformation

Applying these rules to \(g(x)=\frac{1}{x+2}-2\), it implies that the graph of \(f(x)=\frac{1}{x}\) has been shifted 2 units to the left and 2 units down. Thus, instead of the asymptotes being at \(x=0\) and \(y=0\), they will now be at \(x=-2\) and \(y=-2\).

Key Concepts

Function Transformations

To better understand function transformations, let's break down the concept. Function transformation refers to the process of modifying the graph of a function by shifting, stretching, compressing, or reflecting it. These changes occur without altering the function's basic shape.

There are several types of transformations, but for rational functions like \[f(x) = \frac{1}{x}\], we typically focus on shifts. Here's how they generally work:

• Horizontal Shifts: If you add a constant to the variable in the denominator, such as in \(f(x+a) = \frac{1}{x+a}\), the graph will be shifted horizontally. Specifically, it will move to the left by \(a\) units if \(a\) is positive and to the right by \(-a\) units if \(a\) is negative.
• Vertical Shifts: Adding or subtracting a constant to the entire function, such as \(f(x)+a = \frac{1}{x} + a\), results in a vertical shift. The graph will be shifted upwards by \(a\) units if \(a\) is positive and downwards by \(a\) units if \(a\) is negative.

When applying these transformations, always remember that the basic shape stays the same. The only change is the position of the graph on the coordinate plane.

Graphing Rational Functions

Graphing rational functions involves understanding their shapes, transformations, and asymptotes. The hyperbola is a common shape for graphs of rational functions like \[f(x)=\frac{1}{x}\]. When graphing rational functions, follow these steps:

• Identify the Basic Function: For something like \(g(x)=\frac{1}{x+2}-2\), start with the familiar graph of \(f(x)=\frac{1}{x}\).
• Apply Transformations: Using your knowledge of function transformations, shift the graph accordingly. For \(g(x)=\frac{1}{x+2}-2\), you shift the original hyperbola 2 units left and 2 units down.
• Locate Asymptotes: Asymptotes are key to understanding the behavior of rational functions. Identify them for accurate graph placement.

By always starting with the basic graph and applying transformations, you can accurately graph any rational function. Remember that the transformations dictate the new position on the graph axes.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. They are crucial when studying rational functions because they reveal the behavior of the function as it extends towards infinity or negative infinity.

In the case of the function \[g(x)=\frac{1}{x+2}-2\], the asymptotes are shifted from the parent function \(f(x)=\frac{1}{x}\) due to the transformations applied. Here's what you need to know:

• Vertical Asymptotes: These occur when the denominator of the rational function approaches zero. For \(g(x)=\frac{1}{x+2}-2\), the vertical asymptote is at \(x = -2\). This change is due to the horizontal shift of the function.
• Horizontal Asymptotes: For rational functions that have a constant subtracted or added to them, like \(g(x)=\frac{1}{x+2}-2\), the horizontal asymptote is influenced by the transformation. In this case, the horizontal asymptote is at \(y = -2\).

By understanding the shifts and transformations in a rational function, locating asymptotes becomes straightforward. These invisible boundaries inform us of the graph's behavior, allowing for a clearer picture of the function overall.