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.