91Ó°ÊÓ

Transformations help us understand how to modify a base function to get a new one. In the case of the rational function \( g(x)=\frac{1}{x-1} \), we start by identifying the base function \( f(x)=\frac{1}{x} \). This function is a hyperbola centered at the origin. To transform it, we look at the expression inside the function. Here, the change from \( f(x) = \frac{1}{x} \) to \( g(x)=\frac{1}{x-1} \) indicates a horizontal shift.

Remember, the basic transformation rules are:

• A shift to the right: replace \( x \) with \( x-c \) where \( c \) is positive
• A shift to the left: replace \( x \) with \( x+c \)
• A vertical shift: adjust the whole function by adding or subtracting a number after the fraction

In this case, \( x-1 \) means we move everything one unit to the right. No numbers were added or subtracted outside the fraction, so there’s no vertical shift. This keeps the transformation purely horizontal.

Asymptotes are invisible lines that the graph of a function gets close to, but never actually touches. Knowing where these are can make graphing functions much easier. For \( g(x)=\frac{1}{x-1} \), we must find the new positions of the asymptotes after the transformation.

Start by considering the original function \(f(x)=\frac{1}{x}\):

• Vertical asymptote: originally at \(x=0\)
• Horizontal asymptote: at \(y=0\)

When we introduce a horizontal shift by replacing \(x\) with \(x-1\), the vertical asymptote moves to \(x=1\). This happens because the equation \( x-1=0 \) implies \( x=1 \), forming the new boundary where the function is undefined and the graph cannot cross. The horizontal asymptote remains unchanged at \(y=0\) because there's no vertical shift in the transformation. These asymptotes guide us in placing the parts of the hyperbola properly on the graph.

Graphing rational functions like \( g(x)=\frac{1}{x-1} \) involves drawing the asymptotes and understanding the transformation effects visually. Here's how you can do it step-by-step:

• 1. **Draw the Asymptotes:** Begin by sketching the vertical asymptote at \( x=1 \) and the horizontal asymptote at \( y=0 \) on your graph. These will act as barriers that the function will approach but never actually touch.


• 2. **Plot the Hyperbola:** Next, shape the hyperbola so each branch approaches these asymptotes. Use the same mirror-image shape from \( f(x)=\frac{1}{x} \), but remember everything is shifted right by one unit.


• 3. **Check Points:** To ensure accuracy, choose a few points on either side of the vertical asymptote \(x=1\) to substitute into \(g(x)\), confirming the graph's position.

This step-by-step approach ensures that the transformations on the base function are accurately reflected on the graph, giving a precise representation of \(g(x)=\frac{1}{x-1}\). Remember, practice will make recognizing patterns and transformations easier, leading to faster and more accurate graphing.