Please find the rational function for the following: horizontal asymptote is y=6, vertical asymptotes are x=1 and x=5, x intercepts are (-5,0), (1/2, 0), and (6, 0). Y intercept is (0,-18)

To construct a rational function \( f(x) \) with the given properties, we follow these steps:

1. **Horizontal Asymptote \( y = 6 \):** This suggests that the degrees of the numerator and denominator are the same, and the leading coefficient of the numerator is 6 times the leading coefficient of the denominator.

2. **Vertical Asymptotes at \( x = 1 \) and \( x = 5 \):** These indicate that the denominator has factors \( (x - 1) \) and \( (x - 5) \).

3. **X-intercepts at \( x = -5 \), \( x = \frac{1}{2} \), and \( x = 6 \):** These indicate that the numerator has factors \( (x + 5) \), \( (2x - 1) \), and \( (x - 6) \).

4. **Y-intercept at \( (0, -18) \):** This provides a specific value to adjust the constant multiple for the rational function.

Let's define the rational function \( f(x) \) as:
\[ f(x) = \frac{6(x + 5)(2x - 1)(x - 6)}{(x - 1)(x - 5)} \]

To ensure that \( f(0) = -18 \), we need to find the appropriate constant \( k \) such that:
\[ f(0) = k \cdot \frac{6(0 + 5)(2 \cdot 0 - 1)(0 - 6)}{(0 - 1)(0 - 5)} = -18 \]

Simplifying the expression:
\[ f(0) = k \cdot \frac{6(5)(-1)(-6)}{(-1)(-5)} = k \cdot \frac{6 \cdot 5 \cdot 6}{5} = k \cdot 36 \]
\[ 36k = -18 \]
\[ k = -\frac{1}{2} \]

Thus, the rational function becomes:
\[ f(x) = -\frac{1}{2} \cdot \frac{6(x + 5)(2x - 1)(x - 6)}{(x - 1)(x - 5)} \]

Simplifying the constant:
\[ f(x) = \frac{-3(x + 5)(2x - 1)(x - 6)}{(x - 1)(x - 5)} \]

So, the rational function is:
\[ f(x) = \frac{-3(x + 5)(2x - 1)(x - 6)}{(x - 1)(x - 5)} \]

This function satisfies all the given conditions.

Would you like more details or have any questions?

Here are five questions you might ask:
1. How do you determine the horizontal asymptote of a rational function?
2. What happens to the rational function if the degrees of the numerator and denominator are different?
3. Can you explain the significance of vertical asymptotes in a rational function?
4. How do you find the x-intercepts of a rational function?
5. What is the process for finding the y-intercept of a rational function?

**Tip:** When constructing rational functions, ensure that the degrees of the numerator and denominator align with the given asymptotes, and adjust the coefficients to fit specific intercept conditions.

Learn how to construct a rational function with given horizontal and vertical asymptotes, x-intercepts, and y-intercept. This detailed solution covers the steps to create a function that satisfies all specified conditions. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation