Question

problem
$y = \frac{2 - x}{x + 2}$
x-intercept:
e.g., 1
degree:
vertical asymptote:
degree:
leading term:
horizontal asymptote:
pattern of signs:

Explanation:

Step1: Find x-intercept (set y=0)

Set $y=0$, so $\frac{2-x}{x+2}=0$. A rational function is 0 when numerator=0 (denominator≠0):
$2-x=0 \implies x=2$

Step2: Find degree of numerator

Numerator: $2-x = -x+2$, highest power of $x$ is 1, so degree=1.

Step3: Find vertical asymptote (denominator=0)

Set $x+2=0 \implies x=-2$

Step4: Find degree of denominator

Denominator: $x+2$, highest power of $x$ is 1, so degree=1.

Step5: Find leading term

Rewrite function: $y=\frac{-x+2}{x+2}$. Leading terms are highest degree terms of numerator/denominator: $\frac{-x}{x}=-1$ (or $\frac{-x}{x}$ as the leading term ratio)

Step6: Find horizontal asymptote

Since degrees of numerator/denominator are equal, horizontal asymptote is ratio of leading coefficients: $y=\frac{-1}{1}=-1$

Step7: Find sign pattern

Test intervals:

1. $x < -2$: numerator $2-x>0$, denominator $x+2<0$, so $y<0$
2. $-2 < x < 2$: numerator $2-x>0$, denominator $x+2>0$, so $y>0$
3. $x > 2$: numerator $2-x<0$, denominator $x+2>0$, so $y<0$

Sign pattern: Negative, Positive, Negative

Answer:

x-intercept: $2$
degree (numerator): $1$
vertical asymptote: $x=-2$
degree (denominator): $1$
leading term: $\frac{-x}{x}$ (or $-1$)
horizontal asymptote: $y=-1$
pattern of signs: Negative, Positive, Negative