Write an equation for a rational function with:

Vertical asymptotes of x = -4 and x = 6

x intercepts at (3,0) and (-6,0)

y intercept at (0,5)

Use y as the output variable. You may leave your answer in factored form.

To construct a rational function \( y = \frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator, let's use the given conditions:

### Step 1: Vertical Asymptotes
The vertical asymptotes are at \( x = -4 \) and \( x = 6 \). These asymptotes occur where the denominator equals zero. Therefore, the factors in the denominator are:
\[
D(x) = (x + 4)(x - 6)
\]

### Step 2: x-Intercepts
The x-intercepts are at \( (3, 0) \) and \( (-6, 0) \). These intercepts occur where the numerator equals zero. Therefore, the factors in the numerator are:
\[
N(x) = (x - 3)(x + 6)
\]

### Step 3: y-Intercept
The y-intercept is at \( (0, 5) \), which means when \( x = 0 \), \( y = 5 \). To satisfy this, we need to find the constant \( k \) such that:
\[
y = k \cdot \frac{(x - 3)(x + 6)}{(x + 4)(x - 6)}
\]
Substitute \( x = 0 \) and \( y = 5 \):
\[
5 = k \cdot \frac{(0 - 3)(0 + 6)}{(0 + 4)(0 - 6)}
\]
\[
5 = k \cdot \frac{(-3)(6)}{(4)(-6)}
\]
\[
5 = k \cdot \frac{-18}{-24} = k \cdot \frac{3}{4}
\]
\[
k = \frac{5 \cdot 4}{3} = \frac{20}{3}
\]

### Step 4: Final Equation
Now, substitute \( k = \frac{20}{3} \):
\[
y = \frac{20}{3} \cdot \frac{(x - 3)(x + 6)}{(x + 4)(x - 6)}
\]

Thus, the equation of the rational function is:
\[
y = \frac{20(x - 3)(x + 6)}{3(x + 4)(x - 6)}
\]

Would you like more details on how to analyze this function's behavior, or do you have any questions?

### Related Questions:
1. How do you find horizontal asymptotes for rational functions?
2. What happens to a rational function's graph near a vertical asymptote?
3. How can you determine if a rational function has a hole?
4. What role does the leading coefficient play in determining end behavior?
5. How do you sketch the graph of a rational function using its intercepts and asymptotes?

### Tip:
When finding vertical asymptotes, remember that they occur at values of \( x \) that make the denominator zero, provided those values don't also make the numerator zero (which would create a hole instead).

Write an equation for a rational function with: Vertical asymptotes of x = -4 and x = 6; x intercepts at (3,0) and (-6,0); y intercept at (0,5). Use y as the output variable. You may leave your answer in factored form.

Learn how to write a rational function with vertical asymptotes at x = -4 and x = 6, x-intercepts at (3, 0) and (-6, 0), and a y-intercept at (0, 5). This problem covers the key concepts of rational functions, intercepts, and asymptotes. Suitable for high school students in Grades 10-12.

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