To find the coordinates of the point where the asymptotes of the graph $y = \frac{3x + 1}{x - 4}$ intersect, let's examine the vertical and horizontal asymptotes of this rational function.

Step 1: Identify the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero. Set the denominator $x - 4 = 0$ and solve for $x$: $x = 4$ Thus, there is a vertical asymptote at $x = 4$.

Step 2: Identify the Horizontal Asymptote

For rational functions, the horizontal asymptote is determined by the degrees of the numerator and the denominator. Since the degrees of both the numerator ($3x + 1$, degree 1) and the denominator ($x - 4$, degree 1) are the same, we can find the horizontal asymptote by dividing the leading coefficients: $y = \frac{3}{1} = 3$ So, the horizontal asymptote is at $y = 3$.

Step 3: Determine the Intersection Point of the Asymptotes

The intersection point of the vertical asymptote $x = 4$ and the horizontal asymptote $y = 3$ is the point $(4, 3)$.

Answer

The coordinates of the point where the asymptotes intersect are: $(4, 3)$

Would you like further details on this solution or have any questions?

Here are some related questions that expand on this topic:

1. How do you determine the slant asymptote of a rational function when the degree of the numerator is one higher than the degree of the denominator?
2. What happens to the behavior of the function near the asymptotes?
3. Can a rational function have both vertical and oblique asymptotes?
4. How do you find the coordinates of intersection between a curve and its asymptotes if they intersect?
5. How do transformations of the function $y = \frac{1}{x}$ affect its asymptotes?

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