14. What is the vertex of \( 4 y-12=\frac{1}{3 x-6} \)

To find the vertex of the equation \( 4y - 12 = \frac{1}{3x - 6} \), we first need to rewrite it in a more standard form. Starting with the given equation: \[ 4y - 12 = \frac{1}{3x - 6} \] We can isolate \( y \) by adding 12 to both sides: \[ 4y = \frac{1}{3x - 6} + 12 \] Next, we divide everything by 4: \[ y = \frac{1}{4(3x - 6)} + 3 \] Now, we can rewrite the equation in a more recognizable form. The term \( \frac{1}{4(3x - 6)} \) can be simplified: \[ y = \frac{1}{12x - 24} + 3 \] This equation represents a rational function, and it does not have a vertex in the traditional sense as a parabola does. However, we can analyze the behavior of the function. To find the critical points, we can look for the vertical asymptote, which occurs when the denominator is zero: \[ 12x - 24 = 0 \implies 12x = 24 \implies x = 2 \] At \( x = 2 \), the function is undefined, indicating a vertical asymptote. To find the horizontal asymptote, we can analyze the behavior of \( y \) as \( x \) approaches infinity. As \( x \to \infty \), the term \( \frac{1}{12x - 24} \) approaches 0, so: \[ y \to 3 \] Thus, the horizontal asymptote is \( y = 3 \). In summary, the function does not have a vertex like a parabola, but it has a vertical asymptote at \( x = 2 \) and a horizontal asymptote at \( y = 3 \). If you are looking for a specific point of interest, please clarify, and I can assist further!