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The problem asks you to graph \( f(x) = \frac{4x - 1}{2x + 1} \) using transformations based on the graph of \( f(x) = \frac{1}{x} \). To solve this, let's break down the transformation process step by step:

### Step 1: Recognizing the Base Graph
The graph of \( f(x) = \frac{1}{x} \) is a hyperbola with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \), respectively. This is the reference graph.

### Step 2: Analyzing the Transformation
The function we need to graph is \( f(x) = \frac{4x - 1}{2x + 1} \). This is a rational function, and transformations can be used to express how this graph relates to \( f(x) = \frac{1}{x} \). 

Let's simplify the function by rewriting it:

\[
f(x) = \frac{4x - 1}{2x + 1}
\]

#### Vertical Asymptote:
Set the denominator equal to zero to find the vertical asymptote:

\[
2x + 1 = 0 \implies x = -\frac{1}{2}
\]

So the vertical asymptote is at \( x = -\frac{1}{2} \).

#### Horizontal Asymptote:
For large values of \( x \), the degrees of the numerator and denominator are the same, so the horizontal asymptote is determined by the ratio of the leading coefficients:

\[
\lim_{x \to \infty} \frac{4x - 1}{2x + 1} = \frac{4}{2} = 2
\]

Thus, the horizontal asymptote is \( y = 2 \).

### Step 3: Identifying Other Transformations
- The vertical asymptote has shifted from \( x = 0 \) to \( x = -\frac{1}{2} \).
- The horizontal asymptote has shifted from \( y = 0 \) to \( y = 2 \).
- The graph might have a reflection or compression/stretching, but we will also note the shape stays roughly hyperbolic due to the form of the rational function.

### Step 4: Graphing the Transformation
- **Vertical asymptote**: Shift the vertical asymptote from \( x = 0 \) to \( x = -\frac{1}{2} \).
- **Horizontal asymptote**: Shift the horizontal asymptote from \( y = 0 \) to \( y = 2 \).
- **Shape**: The general shape will resemble a hyperbola but will approach the new asymptotes.

This completes the transformation from the basic \( f(x) = \frac{1}{x} \) graph to the graph of \( f(x) = \frac{4x - 1}{2x + 1} \).

Would you like more details on any of the transformations or how to graph this step by step?

Here are some related questions:
1. What role do vertical and horizontal asymptotes play in graphing rational functions?
2. How do you determine the intercepts of a rational function like \( f(x) = \frac{4x - 1}{2x + 1} \)?
3. What types of transformations affect the graph of \( \frac{1}{x} \)?
4. How can you confirm that the graph of \( f(x) = \frac{4x - 1}{2x + 1} \) follows the expected behavior?
5. Can you find any points of intersection between the graph and the axes?

**Tip**: Always check for both vertical and horizontal asymptotes when graphing rational functions to understand their end behavior.

Given the graph of f(x) = 1/x, use transformation to graph f(x) = (4x - 1) / (2x + 1).

Explore how to graph the rational function f(x) = (4x - 1) / (2x + 1) using transformations from the base function f(x) = 1/x. Learn about vertical and horizontal asymptotes, shifts, and the overall shape of the graph. Suitable for Grades 10-12.

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