Alright! Let's go through the steps to graph the rational function [tex]\( f(x) = \frac{2x - 6}{-x + 4} \)[/tex].
### Step 1: Find the Vertical Asymptote
A vertical asymptote occurs where the denominator is equal to zero (since the function goes to infinity or negative infinity).
Set the denominator equal to zero:
[tex]\[ -x + 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So the vertical asymptote is at [tex]\( x = 4 \)[/tex].
### Step 2: Find the Horizontal Asymptote
To find the horizontal asymptote, compare the degrees of the polynomials in the numerator and the denominator.
- The degree of the numerator [tex]\( 2x - 6 \)[/tex] is 1.
- The degree of the denominator [tex]\( -x + 4 \)[/tex] is 1.
Since the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by dividing the leading coefficients:
[tex]\[ \text{Leading coefficient of the numerator} = 2 \][/tex]
[tex]\[ \text{Leading coefficient of the denominator} = -1 \][/tex]
So the horizontal asymptote is:
[tex]\[ y = \frac{2}{-1} = -2 \][/tex]
### Step 3: Plot Points
Let's choose two points on each side of the vertical asymptote to get a clearer picture of the graph.
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \frac{2(1) - 6}{-1 + 4} = \frac{2 - 6}{3} = \frac{-4}{3} \approx -1.33 \][/tex]
2. For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{2(3) - 6}{-3 + 4} = \frac{6 - 6}{1} = \frac{0}{1} = 0 \][/tex]
3. For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{2(5) - 6}{-5 + 4} = \frac{10 - 6}{-1} = \frac{4}{-1} = -4 \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = \frac{2(6) - 6}{-6 + 4} = \frac{12 - 6}{-2} = \frac{6}{-2} = -3 \][/tex]
### Step 4: Sketch the Graph
Finally, put all this information together.
- Draw a vertical dashed line at [tex]\( x = 4 \)[/tex] to represent the vertical asymptote.
- Draw a horizontal dashed line at [tex]\( y = -2 \)[/tex] to represent the horizontal asymptote.
- Plot the points you calculated and draw the curve that approaches the asymptotes:
- Point: (1, -1.33)
- Point: (3, 0)
- Point: (5, -4)
- Point: (6, -3)
Connect these points with a smooth curve, making sure the graph approaches the asymptotes as [tex]\( x \)[/tex] gets very large or very small.
### Conclusion
The graph of [tex]\( f(x) = \frac{2x - 6}{-x + 4} \)[/tex] should clearly show the vertical asymptote at [tex]\( x = 4 \)[/tex] and the horizontal asymptote at [tex]\( y = -2 \)[/tex], along with the plotted points. The curve on each side of the vertical asymptote should approach these asymptotes but never touch them.
