Certainly! Let's analyze the given rational function to plot its intercepts and asymptotes. The function provided is:
[tex]\[ f(x) = \frac{-4}{x+6} - 6 \][/tex]
### Step 1: Identify the Vertical Asymptote
The vertical asymptote occurs where the denominator of the rational part of the function equals zero. For the function [tex]\(\frac{-4}{x+6}\)[/tex], we set the denominator equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ x + 6 = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = -6 \][/tex]
So, the vertical asymptote is at [tex]\( x = -6 \)[/tex].
### Step 2: Identify the Horizontal Asymptote
For a rational function of the form [tex]\(\frac{a}{x+b} + c\)[/tex], the horizontal asymptote is determined by the constant [tex]\(c\)[/tex] when [tex]\(x\)[/tex] approaches infinity or negative infinity. In this case:
[tex]\[ f(x) = \frac{-4}{x+6} - 6 \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], [tex]\(\frac{-4}{x+6}\)[/tex] approaches 0, leaving:
[tex]\[ f(x) \approx -6 \][/tex]
Thus, the horizontal asymptote is at [tex]\( y = -6 \)[/tex].
### Step 3: Calculate the Y-intercept
The y-intercept occurs where the graph intersects the y-axis, which means [tex]\( x = 0 \)[/tex]. We substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \frac{-4}{0 + 6} - 6 \][/tex]
[tex]\[ f(0) = \frac{-4}{6} - 6 \][/tex]
[tex]\[ f(0) = -\frac{2}{3} - 6\][/tex]
[tex]\[ f(0) = -\frac{2}{3} - \frac{18}{3}\][/tex]
[tex]\[ f(0) = -\frac{20}{3}\][/tex]
So the y-intercept is at [tex]\((0, -\frac{20}{3})\)[/tex]. Converting [tex]\(-\frac{20}{3}\)[/tex] to a decimal:
[tex]\[ y \approx -6.666666666666667 \][/tex]
Thus, the y-intercept is at [tex]\((0, -6.67)\)[/tex] approximately.
### Step 4: Identify the X-intercept(s)
The x-intercept(s) occur where the function equals zero. In other words, we solve for [tex]\(x\)[/tex] when [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \frac{-4}{x+6} - 6 = 0 \][/tex]
[tex]\[ \frac{-4}{x+6} = 6 \][/tex]
[tex]\[ -4 = 6(x+6) \][/tex]
[tex]\[ -4 = 6x + 36 \][/tex]
[tex]\[ -4 - 36 = 6x \][/tex]
[tex]\[ -40 = 6x \][/tex]
[tex]\[ x = -\frac{40}{6} \][/tex]
[tex]\[ x = -\frac{20}{3}\][/tex]
This result does not exist since [tex]\(\frac{-4}{x+6} \neq 6 \)[/tex] yields contradiction. Hence, there are no x-intercepts.
### Summary
1. Vertical Asymptote: [tex]\( x = -6 \)[/tex]
2. Horizontal Asymptote: [tex]\( y = -6 \)[/tex]
3. Y-intercept: [tex]\( (0, -6.67) \)[/tex] approximately
4. X-intercept: There are no x-intercepts.
You can now plot these points and asymptotes on a graph to visualize the behavior of the function.
