Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \(g(x)=\frac{1}{x-2}\) is the result of applying a horizontal shift of 2 units to the right to the function \(f(x)=\frac{1}{x}\). To draw it, sketch the base function and then shift the curves 2 units to the right.

Step 1: Identify the Base Function

By analyzing the given function \(g(x)=\frac{1}{x-2}\), it can be observed that it is a transformation of the base function \(f(x)=\frac{1}{x}\). The transformation is \(f(x-2)\), which results in a horizontal shift.

Step 2: Identify the Transformations

The transformation \(f(x-2)\) suggests that the function has undergone a horizontal shift of 2 units to the right. This implies that every point \(x\) on the original function has been shifted to \(x+2\). There are no other transformations as there is no other operation besides the subtraction of 2.

Step 3: Draw the Transformed Graph

Start by drawing the reciprocal function \(f(x)=\frac{1}{x}\), which is a hyperbola whose two separate sections sit in quadrant I and III, getting ever closer but never reaching the x-axis or y-axis. Then, shift every point on this graph 2 units to the right to get the graph of this function. This will result in a new graph similar to the original but translated horizontally.

Key Concepts

Transformations of Functions

Understanding transformations of functions is crucial when it comes to graphing rational functions. Imagine functions as shapes that can be shifted, stretched, or flipped in different directions on a coordinate plane.
In the case of rational functions, such as the hyperbola represented by the reciprocal function, these transformations can dramatically change the graph's appearance. Two types of basic transformations are shifts (both horizontal and vertical) and scalings (which include stretches and compressions).
When graphing, you can start with the base function and systematically apply each transformation. In our example, the base function is the simple reciprocal, \( f(x)=\frac{1}{x} \), which has a characteristic hyperbolic shape. By applying the transformation, we modify this graph step by step to reach our function \( g(x) \).

Importance of Sequential Transformations

Understanding the sequence of transformations is essential. The order can affect the final graph. For example, if you were to apply a horizontal stretch before a horizontal shift, the amount of shifting would differ compared to if you did it in the reverse order.
Always remember to perform transformations in the correct sequence as indicated by the given function. In our exercise example, there is only a horizontal shift to consider, making it a simpler case.

Horizontal Shift

A horizontal shift is a type of transformation that moves a graph left or right across the coordinate plane. To visualize it, just think of sliding a piece of paper horizontally on a table.
In the function \( g(x)=\frac{1}{x-2} \) from our exercise, the '-2' inside the parentheses indicates a shift to the right by 2 units. To be more precise, every \( x \) value of points on the graph of the original function \( f(x) \) has been increased by 2. This is because subtracting a positive number inside the function's formula affects the \( x \) inputs oppositely—moving the graph in the positive direction.

Visualizing Horizontal Shift

A helpful tip for visualizing a horizontal shift is to focus on key points or features of the graph, like the vertex of a parabolic curve or the asymptotes of a hyperbolic curve. For the reciprocal function \( f(x)=\frac{1}{x} \), look at where the curve approaches the axes and move these features two spaces to the right.
Remember that horizontal shifts do not alter the shape of the graph; they only reposition it along the x-axis.

Reciprocal Function

The reciprocal function, typically represented as \( f(x)=\frac{1}{x} \), plays a foundational role in rational functions and their graphs. It's a classic example of a hyperbola, with two separate branches located in the first and third quadrants of a coordinate plane.
These branches exhibit interesting behavior: they get closer and closer to the axes (referred to as 'approaching the asymptotes'), but never actually touch them. This aspect of the graph is crucial when applying transformations because it dictates how the function behaves as \( x \) approaches infinity or zero.

Asymptotic Behavior

A key characteristic of the reciprocal function is its asymptotes—lines that the graph approaches but never crosses. The x-axis and y-axis themselves serve as asymptotes for \( f(x)=\frac{1}{x} \).The reciprocal function's shape is retained in any rational function that can be derived from it through transformations. Thus, when graphing transformations of a reciprocal function, start with its basic shape and thoughtfully apply transformations to predict the new graph's layout accurately.