Question

graph the rational function.
$f(x) = \frac{x - 5}{-2x + 1}$
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.

Explanation:

Step1: Find vertical asymptote

Set denominator to 0: $-2x + 1 = 0$
Solve for $x$: $-2x = -1 \implies x = \frac{1}{2}$

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator (both degree 1). The horizontal asymptote is the ratio of leading coefficients: $y = \frac{1}{-2} = -\frac{1}{2}$

Step3: Find points left of $x=\frac{1}{2}$

Choose $x=0$: $f(0) = \frac{0-5}{-2(0)+1} = \frac{-5}{1} = -5$ → Point: $(0, -5)$
Choose $x=-1$: $f(-1) = \frac{-1-5}{-2(-1)+1} = \frac{-6}{3} = -2$ → Point: $(-1, -2)$

Step4: Find points right of $x=\frac{1}{2}$

Choose $x=1$: $f(1) = \frac{1-5}{-2(1)+1} = \frac{-4}{-1} = 4$ → Point: $(1, 4)$
Choose $x=2$: $f(2) = \frac{2-5}{-2(2)+1} = \frac{-3}{-3} = 1$ → Point: $(2, 1)$

Answer:

1. Vertical Asymptote: $x=\frac{1}{2}$ (dashed vertical line)
2. Horizontal Asymptote: $y=-\frac{1}{2}$ (dashed horizontal line)
3. Plotted points: $(0, -5)$, $(-1, -2)$, $(1, 4)$, $(2, 1)$
4. The graph is two hyperbola branches approaching the asymptotes, passing through the plotted points.