Question

f(x) = \frac{-3x^{2} + 15x}{x^{2} - 7x + 10} 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.

Explanation:

Step1: Factor numerator and denominator

Numerator: $-3x^2 +15x = -3x(x-5)$
Denominator: $x^2 -7x +10 = (x-2)(x-5)$

Step2: Identify holes and simplify

Cancel common factor $(x-5)$ (hole at $x=5$). Simplified function: $f(x)=\frac{-3x}{x-2}$ for $x
eq5$

Step3: Find vertical asymptote

Set simplified denominator to 0: $x-2=0 \implies x=2$

Step4: Find horizontal asymptote

Compare degrees: equal, so asymptote is $y=\frac{-3}{1}=-3$

Step5: Calculate hole coordinates

Substitute $x=5$ into simplified function: $y=\frac{-3(5)}{5-2}=\frac{-15}{3}=-5$. Hole at $(5,-5)$

Step6: Plot points for each piece

For $x<2$: choose $x=0$: $y=\frac{0}{-2}=0$; $x=1$: $y=\frac{-3}{-1}=3$
For $x>2, x
eq5$: choose $x=3$: $y=\frac{-9}{1}=-9$; $x=6$: $y=\frac{-18}{4}=-4.5$

Answer:

1. Draw vertical dashed asymptote at $x=2$, horizontal dashed asymptote at $y=-3$
2. Plot hollow dot at $(5, -5)$
3. Plot solid points: $(0,0)$, $(1,3)$ (left of $x=2$); $(3,-9)$, $(6,-4.5)$ (right of $x=2$)
4. Draw the two branches of $f(x)=\frac{-3x}{x-2}$ approaching the asymptotes, avoiding the hole at $(5,-5)$