Write the equation of a function with vertical asymptotes at x=-3 and x=4, a y intercept at (0,0.4), and an x intercept at (2,0)

To construct the equation of the function with the given characteristics, we can follow these steps:

### 1. **Vertical Asymptotes at \( x = -3 \) and \( x = 4 \)**:
   - Vertical asymptotes generally occur where the denominator of a rational function is zero. To have asymptotes at \( x = -3 \) and \( x = 4 \), the denominator must include factors of \( (x + 3) \) and \( (x - 4) \).

   So, the denominator is:
   \[
   (x + 3)(x - 4)
   \]

### 2. **x-intercept at \( (2, 0) \)**:
   - The x-intercept occurs when \( y = 0 \), meaning the numerator of the function should be zero at \( x = 2 \). Thus, the numerator must have a factor of \( (x - 2) \).

   So, the numerator is:
   \[
   (x - 2)
   \]

### 3. **y-intercept at \( (0, 0.4) \)**:
   - To satisfy this condition, we need to include a constant multiplier \( A \) to adjust the value of the function when \( x = 0 \). We'll determine \( A \) by substituting \( x = 0 \) into the function.

   The form of the function at this point is:
   \[
   y = A \frac{x - 2}{(x + 3)(x - 4)}
   \]

   Substitute \( x = 0 \) and set \( y = 0.4 \):
   \[
   0.4 = A \frac{0 - 2}{(0 + 3)(0 - 4)} = A \frac{-2}{(3)(-4)} = A \frac{-2}{-12} = A \frac{1}{6}
   \]

   Solving for \( A \):
   \[
   A = 0.4 \times 6 = 2.4
   \]

### 4. **Final Equation**:
   Now that we know \( A = 2.4 \), the equation of the function is:
   \[
   y = 2.4 \frac{x - 2}{(x + 3)(x - 4)}
   \]

This is the equation of a rational function that satisfies the given conditions.

Would you like more details or have any questions?

### Related Questions:
1. How do you determine the horizontal asymptote of this function?
2. What happens to the function near the vertical asymptotes?
3. How do you find the domain of this function?
4. Can this function have any other x-intercepts?
5. What is the behavior of the function as \( x \to \infty \)?

### Tip:
When identifying vertical asymptotes, look for where the denominator equals zero but the numerator does not.

Learn how to write the equation of a rational function with vertical asymptotes at x = -3 and x = 4, a y-intercept at (0, 0.4), and an x-intercept at (2, 0). This solution explains the step-by-step process, including how to handle asymptotes, intercepts, and solving for constants.

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