Problem 50 - Algebra and Trigonometry

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

To obtain \( g(x) \) from \( f(x) \) , it needs to be horizontally shifted \(2\) units left and vertically shifted \(2\) units down.

Step 1: Identify the Base Function

We identify our base function from \(g(x)\) as \(f(x)=\frac{1}{x}\). Our task is to graph \(g(x)\) using transformations of \(f(x)\).

Step 2: Horizontal Transformation

Looking at our function \(g(x)=\frac{1}{x+2}-2\), we see that '2' is added to 'x', which is a transformation in the negative x-direction. This is because the term inside the function is \(x+2\). As a rule, we perform the transformation inversely i.e., \(x-->x-2\), thus our function shifts 2 units to the left.

Step 3: Vertical Transformation

Next, observe the '-2' outside the function. This tells us there is also a vertical transformation. As it is \('-2'\), our function moves 2 units down.

Step 4: Combining Transformations

Combining the above transformations: the function \(g(x)\) is obtained by taking the function \(f(x)\) , shifting it two units to the left (making it \(f(x+2)\)), and then shifting it two units down (making it \(f(x+2)-2\) ).

Key Concepts

Rational Functions

Understanding rational functions is essential when analyzing transformations in mathematics. A rational function is a ratio of two polynomials. In its simplest form, it can look like

• A single fraction: \( f(x) = \frac{1}{x} \)
• Or a more complex version: \( g(x) = \frac{P(x)}{Q(x)} \)

where \( P(x) \) and \( Q(x) \) are polynomials. In the context of the exercise, the base rational function is represented as \( f(x) = \frac{1}{x} \).
Rational functions have interesting properties such as vertical and horizontal asymptotes, which are lines that the graph of the function approaches but never touches. The transformations applied to these functions, such as horizontal and vertical shifts, directly affect the location of these asymptotes and therefore the shape and position of the graph itself.
The key takeaway is that rational functions are versatile and widely applied in different areas, making their transformations crucial to understand.

Horizontal Transformation

Horizontal transformations involve shifting a graph left or right along the x-axis. For a function like \( g(x) = \frac{1}{x+2} \), we focus on the expression inside the denominator, \( x+2 \). This signifies a horizontal transformation where we need to adjust the base function \( f(x) = \frac{1}{x} \).
In the context of this exercise:

• \( x \to x+2 \) means we shift the graph 2 units to the left.
• This is because a positive value is combined with \( x \), indicating an operation opposite to its sign for the direction.

This operation changes the location of the vertical asymptote from \( x=0 \) to \( x=-2 \), thereby repositioning the entire structure of the graph horizontally.

Vertical Transformation

Vertical transformations modify the position of a graph along the y-axis. In our function \( g(x)=\frac{1}{x+2}-2 \), the term \(-2\) indicates a vertical movement.
Vertical transformations affect the entire function by:

• Decreasing the value output by the function.
• Shifting all y-values, and hence the horizontal asymptote, downward.

In this exercise:

The graph is moved 2 units down along the y-axis.

Initially, the horizontal asymptote of the base function \( f(x)=\frac{1}{x} \) was at \( y=0 \) but with the transformation becomes \( y=-2 \).
This downward shift emphasizes how the function is impacted by vertical adjustments, visualized by the movement of the graph closer to the bottom edge of the coordinate plane.

Graphing Functions

Graphing functions involves plotting the relationship between input (x-values) and output (y-values) on a coordinate system. For rational functions transformed like \( g(x)=\frac{1}{x+2}-2 \), it is crucial to account for both transformations to accurately depict the change from the initial function.
When graphing:

• Start with the base function \( f(x) = \frac{1}{x} \).
• Apply the horizontal transformation by shifting the graph left 2 units.
• Then perform the vertical transformation, moving the graph down 2 units.

Ensure the asymptotes are updated, as they are key features of rational functions: they set boundaries that the curve approaches.
Carefully executing these steps allows you to transform and graph the function reliably. Visualizing these changes provides a concrete understanding of how functions behave under these transformations.