Graph the rational function.
\[
f(x)=\frac{-3 x^{2}+9 x+2}{x-2}
\]

Step 1 ：To find the x-intercepts, set the numerator equal to zero and solve for x: \(-3x^2 + 9x + 2 = 0\).

Step 2 ：This is a quadratic equation, which can be solved using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3 ：Here, \(a = -3\), \(b = 9\), and \(c = 2\). Plugging these values into the quadratic formula gives: \(x = \frac{-9 \pm \sqrt{(9)^2 - 4*(-3)*2}}{2*(-3)}\).

Step 4 ：Solving the equation gives: \(x = \frac{-9 \pm \sqrt{81 + 24}}{-6}\) and \(x = \frac{-9 \pm \sqrt{105}}{-6}\).

Step 5 ：So, the x-intercepts are \(x = 1.5 \pm \frac{\sqrt{105}}{6}\).

Step 6 ：To find the y-intercept, set \(x = 0\) in the function: \(f(0) = \frac{-3*(0)^2 + 9*0 + 2}{0 - 2}\).

Step 7 ：Solving the equation gives: \(f(0) = \frac{2}{-2} = -1\).

Step 8 ：So, the y-intercept is \((0, -1)\).

Step 9 ：To find the vertical asymptote, set the denominator equal to zero and solve for x: \(x - 2 = 0\).

Step 10 ：Solving the equation gives: \(x = 2\).

Step 11 ：So, the vertical asymptote is \(x = 2\).

Step 12 ：To find the horizontal asymptote, compare the degrees of the numerator and denominator. The degree of the numerator is 2 and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Step 13 ：To find any holes in the graph, check if a factor in the numerator and denominator can be cancelled out. In this case, there are no common factors in the numerator and denominator, so there are no holes in the graph.

Step 14 ：Now, we can graph the function using these key features. Plot the x-intercepts, y-intercept, and vertical asymptote. Since there is no horizontal asymptote, the graph will approach infinity or negative infinity as x approaches the vertical asymptote. Since there are no holes in the graph, the graph is continuous except at the vertical asymptote.