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Unproctored Placement Assessment
Question 14
Rob
Graph the rational function.
$$f(x)=\frac{2x-2}{-x+4}$$

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.

Step 1 ：Identify the vertical and horizontal asymptotes. The vertical asymptote is found by setting the denominator equal to zero and solving for x. The horizontal asymptote is determined by the degree of the numerator and the denominator.

Step 2 ：The vertical asymptote is $x=4$ (from $-x+4=0$) and the horizontal asymptote is $y=2$ (from $2/1$, since the degrees of the numerator and denominator are equal).

Step 3 ：Plot two points on each piece of the graph. Choose points that are easy to calculate, such as $x=0$ and $x=1$ for the piece to the left of the vertical asymptote, and $x=5$ and $x=6$ for the piece to the right of the vertical asymptote.

Step 4 ：Calculate the y-values for these points. The y-values for the points $(0,f(0))$, $(1,f(1))$, $(5,f(5))$, and $(6,f(6))$ are $-0.5$, $0.0$, $-8.0$, and $-5.0$ respectively.

Step 5 ：Plot these points along with the vertical and horizontal asymptotes to sketch the graph of the function.

Step 6 ：Final Answer: The vertical asymptote is $\boxed{{\displaystyle x=4}}$, the horizontal asymptote is $\boxed{{\displaystyle y=2}}$, and the points on the graph are $\boxed{{\displaystyle (0,-0.5)}}$, $\boxed{{\displaystyle (1,0.0)}}$, $\boxed{{\displaystyle (5,-8.0)}}$, and $\boxed{{\displaystyle (6,-5.0)}}$.