Questions: Graph the rational function. f(x) = (-2x + 6) / (-x + 4) Start by drawing the vertical and horizontal asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph-a-function button.

Solution Steps

The vertical asymptote occurs where the denominator is equal to zero. \[-x + 4 = 0\] \[x = 4\] So, the vertical asymptote is \(x = 4\).

The horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator when the degrees of the numerator and denominator are the same. \[y = \frac{-2}{-1} = 2\] So, the horizontal asymptote is \(y = 2\).

Let's choose \(x\) values to the left and right of the vertical asymptote \(x=4\).

For \(x = 0\): \[f(0) = \frac{-2(0) + 6}{-0 + 4} = \frac{6}{4} = \frac{3}{2} = 1.5\] So, the point is \((0, 1.5)\).

For \(x = 2\): \[f(2) = \frac{-2(2) + 6}{-2 + 4} = \frac{-4 + 6}{2} = \frac{2}{2} = 1\] So, the point is \((2, 1)\).

For \(x = 5\): \[f(5) = \frac{-2(5) + 6}{-5 + 4} = \frac{-10 + 6}{-1} = \frac{-4}{-1} = 4\] So, the point is \((5, 4)\).

For \(x = 6\): \[f(6) = \frac{-2(6) + 6}{-6 + 4} = \frac{-12 + 6}{-2} = \frac{-6}{-2} = 3\] So, the point is \((6, 3)\).

Plot the asymptotes and the calculated points on the graph. Then, draw the branches of the rational function approaching the asymptotes.

Final Answer