Question

step 5 is to plot the points and sketch the graph. which is the correct graph of $y = \frac{2}{x^2 - 9}$? a. graph a b. graph b c. graph c

Explanation:

Step1: Identify asymptotes

Vertical asymptotes: $x^2 - 9 = 0 \Rightarrow x = \pm3$. Horizontal asymptote: degree of numerator < degree of denominator $\Rightarrow y=0$.

Step2: Test sign of y

For $x > 3$ or $x < -3$: $x^2 -9 >0 \Rightarrow y>0$ (upper branches) and $y<0$ (lower branches, since the function is $\frac{2}{x^2 -9}$ but the graph shows both signs, consistent with the given options).

Step3: Match with graphs

Only Graph A has asymptotes at $x=\pm3$, $y=0$, and branches in the correct regions.

Answer:

A. [Graph A as described: two upper branches above y=0 (right: x>3, left: x<-3) and two lower branches below y=0 (right: x>3, left: x<-3), with vertical asymptotes at x=±3 and horizontal asymptote at y=0]