Question

1. (20 points) graph the rational function. identify all asymptotes, all holes, all zeros and the \\(y\\)-intercept.

\\p(x) = \frac{-3x^2 - 3x + 126}{x^2 - 3x - 10}\\

v.a.:
h.a.:
o.a.:
holes:
zeros:
\\(y\\)-int:

Explanation:

Step1: Factor the numerator and denominator

$$p(x) = \frac{-3(x^2 + x - 42)}{x^2 - 3x - 10} = \frac{-3(x - 6)(x + 7)}{(x - 5)(x + 2)}$$

Step2: Identify vertical asymptotes

$$x - 5 = 0 \implies x = 5, \quad x + 2 = 0 \implies x = -2$$

Step3: Identify horizontal asymptotes

$$y = \frac{-3}{1} = -3$$

Step4: Identify oblique asymptotes

Since degree of numerator equals denominator, there is no oblique asymptote.

Step5: Identify holes

No common factors in numerator and denominator, so there are no holes.

Step6: Identify zeros

$$-3(x - 6)(x + 7) = 0 \implies x = 6, \quad x = -7$$

Step7: Identify y-intercept

$$p(0) = \frac{126}{-10} = -12.6$$

Answer:

V.A.: \(x = -2\)
H.A.: \(y = -3\)
O.A.: None
Holes: \((5, -11)\)
Zeros: \((6, 0)\)
\(y\)-int: \((0, -12.6)\)