Let's take a detailed, step-by-step approach to finding the y-intercept, horizontal asymptote, and vertical asymptote for the function [tex]\( f(x) = \frac{-3}{x + 2} \)[/tex].
### Finding the y-intercept:
The y-intercept occurs where the value of [tex]\( x \)[/tex] is zero. To find the y-intercept, we substitute [tex]\( x = 0 \)[/tex] into the given function and solve for [tex]\( f(x) \)[/tex]:
[tex]\[
f(0) = \frac{-3}{0 + 2} = \frac{-3}{2}
\][/tex]
Thus, the y-intercept is [tex]\(\left(0, -\frac{3}{2}\right)\)[/tex].
### Finding the horizontal asymptote:
The horizontal asymptote represents the value that [tex]\( f(x) \)[/tex] approaches as [tex]\( x \)[/tex] approaches infinity or negative infinity. For a rational function of the form [tex]\( \frac{a}{bx + c} \)[/tex], the horizontal asymptote can be determined by looking at the behavior as [tex]\( x \)[/tex] becomes very large. Here, as [tex]\( x \)[/tex] approaches infinity or negative infinity, the value of the fraction [tex]\(\frac{-3}{x + 2}\)[/tex] approaches zero because the numerator is a constant and the denominator grows without bound.
Therefore, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
### Finding the vertical asymptote:
The vertical asymptote occurs where the denominator of the function equals zero, causing the function to be undefined. To find the vertical asymptote, we set the denominator of the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x + 2 = 0
\][/tex]
Solving this equation, we get:
[tex]\[
x = -2
\][/tex]
Thus, the vertical asymptote is [tex]\( x = -2 \)[/tex].
### Summary:
- y-intercept: [tex]\(\left(0, -\frac{3}{2}\right)\)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- Vertical Asymptote: [tex]\( x = -2 \)[/tex]
