write a rational function with the following characteristics vertical asymptote x = -2, x = 3 horizontal asymptote y = 0 x intercept where x =1

A rational function with the given characteristics can be written as:

\[
f(x) = \frac{k(x - 1)}{(x + 2)(x - 3)}
\]

Where:
- **Vertical asymptotes** at \( x = -2 \) and \( x = 3 \) imply that the denominator has factors \( (x + 2) \) and \( (x - 3) \).
- **Horizontal asymptote** at \( y = 0 \) indicates that the degree of the numerator is less than the degree of the denominator, which is satisfied by this function since the denominator is a quadratic and the numerator is linear.
- **x-intercept** at \( x = 1 \) means that the numerator must have a factor of \( (x - 1) \).

To determine the constant \( k \), you would need additional information, but for this case, the general form of the function is as above.

Would you like more details on how this function satisfies the conditions? 

Here are five related questions to explore further:
1. What is the process for finding vertical asymptotes in rational functions?
2. How does the degree of the numerator and denominator determine horizontal asymptotes?
3. What happens to the function near the vertical asymptotes?
4. How can you determine the behavior of a rational function at its x-intercept?
5. How would the function change if there were a slant asymptote instead of a horizontal asymptote?

**Tip:** Always check the degrees of the numerator and denominator to determine the type of asymptotes a rational function has.

write a rational function with the following characteristics: vertical asymptote x = -2, x = 3, horizontal asymptote y = 0, x-intercept where x = 1

Explore how to construct a rational function with vertical asymptotes at x = -2 and x = 3, a horizontal asymptote at y = 0, and an x-intercept at x = 1. Learn key concepts such as vertical and horizontal asymptotes, and x-intercepts, suitable for students in Grades 9-12.

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