Sure, let's determine the intercepts and asymptotes of the rational function [tex]\( f(x) = -5 - \frac{3}{x+3} \)[/tex].
### Steps:
1. Determine the vertical asymptote:
The vertical asymptote occurs where the denominator is zero.
[tex]\[
x + 3 = 0 \rightarrow x = -3
\][/tex]
So, the vertical asymptote is at [tex]\( x = -3 \)[/tex].
2. Determine the horizontal asymptote:
For the horizontal asymptote of a rational function where the degree of the numerator is less than the degree of the denominator, we look at the constant term since [tex]\( f(x) = -5 - \frac{3}{x+3} \)[/tex].
As [tex]\( x \)[/tex] approaches infinity, [tex]\( \frac{3}{x+3} \)[/tex] approaches 0. Thus the horizontal asymptote is:
[tex]\[
y = -5
\][/tex]
3. Find the y-intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex].
[tex]\[
f(0) = -5 - \frac{3}{0+3} = -5 - \frac{3}{3} = -5 - 1 = -6
\][/tex]
So, the y-intercept is [tex]\( (0, -6) \)[/tex].
4. Find the x-intercept:
The x-intercept occurs when [tex]\( f(x) = 0 \)[/tex]. Set [tex]\( f(x) \)[/tex] to 0 and solve for [tex]\( x \)[/tex].
[tex]\[
0 = -5 - \frac{3}{x+3}
\][/tex]
[tex]\[
5 = -\frac{3}{x+3}
\][/tex]
[tex]\[
-5 = \frac{3}{x+3}
\][/tex]
[tex]\[
-5(x+3) = 3
\][/tex]
[tex]\[
-5x - 15 = 3
\][/tex]
[tex]\[
-5x = 18
\][/tex]
[tex]\[
x = -\frac{18}{5} = -3.6
\][/tex]
So, the x-intercept is [tex]\( \left(-3.6, 0\right) \)[/tex].
### Summary of Intercepts and Asymptotes:
- Vertical Asymptote: [tex]\( x = -3 \)[/tex]
- Horizontal Asymptote: [tex]\( y = -5 \)[/tex]
- x-Intercept: [tex]\( \left(-3.6, 0\right) \)[/tex]
- y-Intercept: [tex]\( (0, -6) \)[/tex]
### Table Presentation
| Vertical Asymptote | Horizontal Asymptote |
|--------------------|----------------------|
| [tex]\(x = -3\)[/tex] | [tex]\(y = -5\)[/tex] |
| [tex]$x$[/tex]-Intercept | [tex]$y$[/tex]-Intercept |
|------------------|---------------|
| [tex]\(-3.6, 0\)[/tex] | [tex]\((0, -6)\)[/tex] |
So when plotting, ensure to mark the vertical line at [tex]\( x = -3 \)[/tex], the horizontal line at [tex]\( y = -5 \)[/tex], the point [tex]\(-3.6, 0\)[/tex] for the x-intercept, and [tex]\(0, -6\)[/tex] for the y-intercept.
