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.