To graph the rational function [tex]\( f(x) = \frac{-6}{-x + 6} \)[/tex], we need to determine its vertical and horizontal asymptotes. Let's break it down step-by-step:
### Vertical Asymptote
A vertical asymptote occurs where the denominator of the function equals zero because the function value approaches infinity (or negative infinity) at this point.
1. Identify the denominator: The denominator of [tex]\( f(x) \)[/tex] is [tex]\( -x + 6 \)[/tex].
2. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 6 = 0 \][/tex]
[tex]\[ -x = -6 \][/tex]
[tex]\[ x = 6 \][/tex]
Therefore, the function has a vertical asymptote at [tex]\( x = 6 \)[/tex].
### Horizontal Asymptote
A horizontal asymptote describes the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex].
1. To find the horizontal asymptote for rational functions, observe the degrees of the numerator and the denominator.
- In [tex]\( f(x) = \frac{-6}{-x + 6} \)[/tex], the degree of the numerator (which is a constant, hence degree 0) is less than the degree of the denominator (which is 1).
2. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
Therefore, the function has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Graphing the Function
To graph the function [tex]\( f(x) = \frac{-6}{-x + 6} \)[/tex], use the following steps:
1. Draw the vertical asymptote at [tex]\( x = 6 \)[/tex] as a dashed vertical line.
2. Draw the horizontal asymptote at [tex]\( y = 0 \)[/tex] as a dashed horizontal line.
Next, plot a few points to understand the behavior of the function near these asymptotes and sketch the curve accordingly.
#### Plotting Points
Select points on either side of the vertical asymptote [tex]\( x = 6 \)[/tex] to see the behavior of the function.
For example:
- [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = \frac{-6}{-(4) + 6} = \frac{-6}{-4 + 6} = \frac{-6}{2} = -3 \][/tex]
- [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = \frac{-6}{-(8) + 6} = \frac{-6}{-8 + 6} = \frac{-6}{-2} = 3 \][/tex]
- [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = \frac{-6}{-(7) + 6} = \frac{-6}{-7 + 6} = \frac{-6}{-1} = 6 \][/tex]
- [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{-6}{-(5) + 6} = \frac{-6}{-5 + 6} = \frac{-6}{1} = -6 \][/tex]
#### Sketch the Graph
1. Plot the points calculated above: [tex]\( (4, -3) \)[/tex], [tex]\( (8, 3) \)[/tex], [tex]\( (7, 6) \)[/tex], and [tex]\( (5, -6) \)[/tex].
2. Draw a curve approaching the vertical asymptote [tex]\( x = 6 \)[/tex] ensuring that as [tex]\( x \)[/tex] approaches 6 from the left, [tex]\( f(x) \)[/tex] decreases towards [tex]\( -\infty \)[/tex], and as [tex]\( x \)[/tex] approaches 6 from the right, [tex]\( f(x) \)[/tex] increases towards [tex]\( \infty \)[/tex].
3. Ensure that the function approaches the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \rightarrow \pm \infty \)[/tex].
By following these steps, you can effectively graph the function [tex]\( f(x) = \frac{-6}{-x + 6} \)[/tex] with its corresponding asymptotes.
