graph the rational function.\n f(x)=\frac{-2x^{2}-16x}{x^{2}+8x + 12} \nstart by drawing the vertical and…

Explanation:

Step1: Factor the function

First, factor the numerator and denominator. The numerator (-2x^{2}-16x=-2x(x + 8)). The denominator (x^{2}+8x + 12=(x + 2)(x+6)). So (f(x)=\frac{-2x(x + 8)}{(x + 2)(x + 6)}).

Step2: Find vertical asymptotes

Set the denominator equal to zero ((x + 2)(x + 6)=0). Solving (x+2 = 0) gives (x=-2) and solving (x + 6=0) gives (x=-6). So the vertical asymptotes are (x=-2) and (x=-6).

Step3: Find horizontal asymptote

Since the degree of the numerator and denominator are the same (both degree 2), the horizontal - asymptote is (y=\frac{-2}{1}=-2) (the ratio of the leading coefficients).

Step4: Find x - intercepts

Set the numerator equal to zero (-2x(x + 8)=0). Solving (-2x=0) gives (x = 0) and solving (x+8=0) gives (x=-8). So the x - intercepts are (x = 0) and (x=-8).

Step5: Find y - intercept

Set (x = 0) in the function (f(0)=\frac{-2\times0^{2}-16\times0}{0^{2}+8\times0 + 12}=0). So the y - intercept is (y = 0).

Step6: Plot points

For (x=-7) (left of (x=-6)), (f(-7)=\frac{-2\times(-7)\times(-7 + 8)}{(-7 + 2)\times(-7+6)}=\frac{14}{5}=2.8). For (x=-5) (between (x=-6) and (x=-2)), (f(-5)=\frac{-2\times(-5)\times(-5 + 8)}{(-5 + 2)\times(-5+6)}=\frac{30}{-3}=-10). For (x=-1) (right of (x=-2)), (f(-1)=\frac{-2\times(-1)\times(-1 + 8)}{(-1 + 2)\times(-1+6)}=\frac{14}{5}=2.8).

To graph: Draw the vertical asymptotes (x=-2) and (x=-6) as dashed lines, draw the horizontal asymptote (y=-2) as a dashed line. Plot the x - intercepts ((0,0)) and ((-8,0)), the y - intercept ((0,0)), and the points ((-7,2.8)), ((-5,-10)), ((-1,2.8)) and then sketch the graph of the rational function approaching the asymptotes.

Answer:

Graph the function with vertical asymptotes (x=-2) and (x=-6), horizontal asymptote (y=-2), x - intercepts at (x = 0) and (x=-8), y - intercept at (y = 0) and additional points ((-7,2.8)), ((-5,-10)), ((-1,2.8))