Questions: Graph the rational function. f(x) = (2x^2 - 6x) / (x^2 - 4x + 3) Start by drawing the asymptotes (if there are any). Then plot two points on each piece of the graph. Finally, click on the graph-a-function button. Be sure to plot a hollow dot wherever there is a "hole" in the graph.

Solution Steps

The given rational function is

\[ f(x) = \frac{2x^2 - 6x}{x^2 - 4x + 3} \]

To find the vertical asymptotes, set the denominator equal to zero:

\[ x^2 - 4x + 3 = 0 \]

Factoring the quadratic equation, we get:

\[ (x - 3)(x - 1) = 0 \]

Thus, the vertical asymptotes are at \(x = 3\) and \(x = 1\).

For the horizontal asymptote, compare the degrees of the numerator and the denominator. Both are degree 2, so the horizontal asymptote is the ratio of the leading coefficients:

\[ y = \frac{2}{1} = 2 \]

To find any holes, check for common factors in the numerator and the denominator. The numerator can be factored as:

\[ 2x(x - 3) \]

The common factor is \(x - 3\), so there is a hole at \(x = 3\).

Choose points around the vertical asymptotes and the hole to plot:

1. For \(x < 1\), choose \(x = 0\): \[ f(0) = \frac{2(0)^2 - 6(0)}{(0)^2 - 4(0) + 3} = 0 \]

2. For \(1 < x < 3\), choose \(x = 2\): \[ f(2) = \frac{2(2)^2 - 6(2)}{(2)^2 - 4(2) + 3} = \frac{8 - 12}{4 - 8 + 3} = \frac{-4}{-1} = 4 \]

3. For \(x > 3\), choose \(x = 4\): \[ f(4) = \frac{2(4)^2 - 6(4)}{(4)^2 - 4(4) + 3} = \frac{32 - 24}{16 - 16 + 3} = \frac{8}{3} \]

Final Answer