Find the slope of the graph of the function \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \). Then find an equation for the line tangent to the graph at that point. The slope of the graph of the function \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \) is \( -\frac{3}{2} \) (Type an integer or a simplified fraction.) The equation for the line tangent to \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \) is \( y= \)

To find the slope of the function \( g(x) = \frac{3x}{x-2} \) at the point \( (4,6) \), we can use differentiation. The derivative \( g'(x) \) gives the slope of the tangent line at any point \( x \). After computing \( g'(x) \) using the quotient rule, you would evaluate it at \( x = 4 \) to confirm the slope is indeed \( -\frac{3}{2} \). Now, to find the tangent line's equation, use the point-slope form \( y - y_1 = m(x - x_1) \) where \( m \) is the slope and \( (x_1, y_1) \) is the given point. Plugging in \( m = -\frac{3}{2} \) and the point \( (4, 6) \), the equation becomes: \[ y - 6 = -\frac{3}{2}(x - 4) \] To express it in slope-intercept form, just simplify: \[ y = -\frac{3}{2}x + 6 + 6 \] \[ y = -\frac{3}{2}x + 12 \] So the completed equation of the tangent line is \( y = -\frac{3}{2}x + 12 \). Exploring rational functions like \( g(x) \) can unveil fascinating patterns and behaviors, especially near vertical asymptotes where the function can behave wildly! When analyzing such functions, keep an eye on those moments—they can lead to key insights about limits and continuity. In real-world applications, the concept of tangents isn't just a math exercise; it appears in physics too! For instance, in motion along a curved path, the tangent line at any point gives the instantaneous velocity—super crucial for cars making sharp turns or roller coasters speeding along loops!