Certainly! To graph the rational function [tex]\( f(x) = \frac{-4}{-x+2} \)[/tex], we'll follow these steps:
### 1. Identify the Vertical Asymptote
The vertical asymptote occurs where the denominator is zero.
[tex]\[ -x + 2 = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
### 2. Identify the Horizontal Asymptote
To find the horizontal asymptote, we consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] tends to very large or very small values. For the rational function of the form [tex]\( \frac{a}{bx + c} \)[/tex], the horizontal asymptote is [tex]\( y = 0 \)[/tex].
### 3. Plot the Asymptotes
Draw the vertical asymptote as a dashed line at [tex]\( x = 2 \)[/tex].
Draw the horizontal asymptote as a dashed line at [tex]\( y = 0 \)[/tex].
### 4. Choose Points on Each Side of the Asymptote
We need to choose points on both sides of the vertical asymptote (i.e., for [tex]\( x < 2 \)[/tex] and [tex]\( x > 2 \)[/tex]).
#### For [tex]\( x < 2 \)[/tex]:
- Choose [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{-4}{-0 + 2} = \frac{-4}{2} = -2 \][/tex]
So, the point [tex]\((0, -2)\)[/tex].
- Choose [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = \frac{-4}{-(-4) + 2} = \frac{-4}{4 + 2} = \frac{-4}{6} = -\frac{2}{3} \approx -0.67 \][/tex]
So, the point [tex]\((-4, -0.67)\)[/tex].
#### For [tex]\( x > 2 \)[/tex]:
- Choose [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = \frac{-4}{-4 + 2} = \frac{-4}{-2} = 2 \][/tex]
So, the point [tex]\((4, 2)\)[/tex].
- Choose [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = \frac{-4}{-6 + 2} = \frac{-4}{-4} = 1 \][/tex]
So, the point [tex]\((6, 1)\)[/tex].
### 5. Plot the Points and Sketch the Graph
- Plot the points [tex]\((0, -2)\)[/tex] and [tex]\((-4, -0.67)\)[/tex] on the left side of the vertical asymptote.
- Plot the points [tex]\((4, 2)\)[/tex] and [tex]\((6, 1)\)[/tex] on the right side of the vertical asymptote.
### 6. Draw the Graph
Connect the points smoothly, making sure the graph approaches the vertical asymptote [tex]\( x = 2 \)[/tex] but never touches it, and approaches the horizontal asymptote [tex]\( y = 0 \)[/tex].
### Final Graph
You will have two sections of the graph:
- One section for [tex]\( x < 2 \)[/tex] curving towards the vertical asymptote on the right and towards the horizontal asymptote downwards.
- Another section for [tex]\( x > 2 \)[/tex] curving towards the vertical asymptote on the left and towards the horizontal asymptote upwards.
The final visual should resemble a hyperbola split by the vertical asymptote at [tex]\(x = 2\)[/tex] and approaching the horizontal asymptote [tex]\(y = 0\)[/tex].
