determine the equation of the horizontal asymptope of the graph og the function x-6/2x+1

To determine the equation of the horizontal asymptote for the function \( f(x) = \frac{x - 6}{2x + 1} \), follow these steps:

### Step 1: Identify the degrees of the numerator and denominator.
- The numerator \( x - 6 \) has a degree of 1 (since \( x \) is to the power of 1).
- The denominator \( 2x + 1 \) also has a degree of 1.

### Step 2: Compare the degrees of the numerator and denominator.
- When the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

### Step 3: Find the leading coefficients.
- The leading coefficient of the numerator (the coefficient of \( x \)) is 1.
- The leading coefficient of the denominator is 2.

### Step 4: Determine the horizontal asymptote.
- The horizontal asymptote is \( y = \frac{1}{2} \).

So, the equation of the horizontal asymptote is **\( y = \frac{1}{2} \)**.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. What is the vertical asymptote of the function \( f(x) = \frac{x - 6}{2x + 1} \)?
2. How would you find the x-intercept of the function \( f(x) = \frac{x - 6}{2x + 1} \)?
3. What happens to the horizontal asymptote if the degrees of the numerator and denominator are different?
4. Can a rational function have more than one horizontal asymptote? Why or why not?
5. How does the horizontal asymptote affect the behavior of the function as \( x \) approaches infinity?

**Tip:** Always check the degrees of the numerator and denominator when analyzing horizontal asymptotes in rational functions.

Learn how to find the horizontal asymptote of the rational function f(x) = (x - 6) / (2x + 1) using leading coefficients and degrees. This problem explores rational functions and their asymptotic behavior, suitable for high school students studying algebra and calculus.

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