To graph the rational function [tex]\( f(x) = \frac{-6}{-x+6} \)[/tex], we need to follow a systematic approach. We'll identify the vertical and horizontal asymptotes first, and then plot significant points on the graph.
### Step 1: Identifying Asymptotes
Vertical Asymptote:
A vertical asymptote occurs where the denominator of the function equals zero since the function is undefined at those points. Set the denominator [tex]\(-x + 6 = 0\)[/tex]:
[tex]\[
-x + 6 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
-x = -6 \implies x = 6
\][/tex]
Thus, the vertical asymptote is [tex]\( x = 6 \)[/tex].
Horizontal Asymptote:
A horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. The degrees of the numerator and the denominator are both 1 (linear). For this type of rational function, the horizontal asymptote is given by the ratio of their leading coefficients. In this case, the leading coefficients are [tex]\(-6\)[/tex] (numerator) and [tex]\(-1\)[/tex] (denominator, from [tex]\(-x+6\)[/tex]):
[tex]\[
\text{Horizontal Asymptote} = \frac{-6}{-1} = 6
\][/tex]
Thus, the horizontal asymptote is [tex]\( y = 6 \)[/tex].
### Step 2: Plotting Points
Next, we'll choose two points on either side of the vertical asymptote [tex]\( x = 6 \)[/tex] to better understand the behavior of the function.
Points to the left of [tex]\( x = 6 \)[/tex]:
- Choose [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = \frac{-6}{-(4) + 6} = \frac{-6}{2} = -3
\][/tex]
So, one point is [tex]\((4, -3)\)[/tex].
- Choose [tex]\( x = 5 \)[/tex]:
[tex]\[
f(5) = \frac{-6}{-(5) + 6} = \frac{-6}{1} = -6
\][/tex]
So, another point is [tex]\((5, -6)\)[/tex].
Points to the right of [tex]\( x = 6 \)[/tex]:
- Choose [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = \frac{-6}{-(7) + 6} = \frac{-6}{-1} = 6
\][/tex]
So, one point is [tex]\((7, 6)\)[/tex].
- Choose [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = \frac{-6}{-(8) + 6} = \frac{-6}{-2} = 3
\][/tex]
So, another point is [tex]\((8, 3)\)[/tex].
### Step 3: Drawing the Graph
Now, we can sketch the graph incorporating the asymptotes and points.
1. Draw the Vertical Asymptote: Draw a dashed vertical line at [tex]\( x = 6 \)[/tex].
2. Draw the Horizontal Asymptote: Draw a dashed horizontal line at [tex]\( y = 6 \)[/tex].
Next, plot the points:
- Left of the asymptote: [tex]\((4, -3)\)[/tex] and [tex]\((5, -6)\)[/tex]
- Right of the asymptote: [tex]\((7, 6)\)[/tex] and [tex]\((8, 3)\)[/tex]
Finally, draw smooth curves approaching the asymptotes through the plotted points.
Your graph will show that the function gets closer to [tex]\( y = 6 \)[/tex] as [tex]\( x \)[/tex] goes to positive or negative infinity, and it will approach the vertical asymptote [tex]\( x = 6 \)[/tex] but never touch or cross it.
