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To graph the function \( h(x) = \frac{1}{x - 1} \) using transformations, we start with the basic function \( f(x) = \frac{1}{x} \) and apply transformations accordingly. Here's the step-by-step process:

### Step 1: Identify the Basic Function
- The basic function is \( f(x) = \frac{1}{x} \). This function has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \). It also has symmetry about the origin, and its graph is in the first and third quadrants.

### Step 2: Apply the Horizontal Shift
- The function \( h(x) = \frac{1}{x - 1} \) indicates a horizontal shift of the graph of \( f(x) \) by 1 unit to the right.
- This changes the vertical asymptote from \( x = 0 \) to \( x = 1 \).

### Step 3: Identify the Asymptotes and Plot Key Points
- **Vertical Asymptote**: \( x = 1 \) (from the shift).
- **Horizontal Asymptote**: The horizontal asymptote remains at \( y = 0 \) because the function is still of the form \( \frac{1}{x - 1} \).

### Step 4: Graph the Transformed Function
- The graph will still resemble the basic function's shape, but now it will be centered around \( x = 1 \). 
- As \( x \) approaches 1 from the left, \( h(x) \) approaches \( -\infty \).
- As \( x \) approaches 1 from the right, \( h(x) \) approaches \( \infty \).
- The graph will approach the horizontal asymptote \( y = 0 \) as \( x \) goes to \( \pm \infty \).

### Final Graph
- Plot the asymptote \( x = 1 \) and draw the curve in the first and third quadrants, shifted right by one unit.

Would you like further details or explanations on any part of this process?

### Related Questions
1. How would you graph \( h(x) = \frac{1}{x + 2} \) using transformations?
2. What are the asymptotes of \( g(x) = \frac{1}{x^2 - 4} \)?
3. How does the graph of \( f(x) = \frac{1}{x} \) change when multiplied by a negative coefficient?
4. What effect does shifting \( f(x) = \frac{1}{x} \) vertically have on its graph?
5. Can you explain how to find the domain and range of \( h(x) = \frac{1}{x - 1} \)?

### Tip
When graphing rational functions, always start by identifying asymptotes as they provide a framework for sketching the curve accurately.

Use transformations of f(x) = 1/x or f(x) = 1/x^2 to graph the rational function h(x) = 1/(x - 1).

Learn how to graph the rational function h(x) = 1/(x - 1) using transformations. This step-by-step guide covers key concepts like horizontal shifts, vertical asymptotes, and graphing rational functions. Ideal for high school students in grades 9-12.

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