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).