Answer :
To analyze and graph the function [tex]\( f(x) = \frac{2x - 7}{-x + 2} \)[/tex], we will break it down into steps and identify the key features first.
### Step 1: Find the Vertical Asymptote
Vertical asymptotes occur where the denominator of the function is zero and the numerator is not zero. This is because division by zero is undefined.
Given the function:
[tex]\[ f(x) = \frac{2x - 7}{-x + 2} \][/tex]
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 2 = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
### Step 2: Find the Horizontal Asymptote
Horizontal asymptotes describe the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex] or [tex]\( -\infty \)[/tex].
We examine the degrees of the numerator and the denominator:
- The degree of the numerator [tex]\( 2x-7 \)[/tex] is 1.
- The degree of the denominator [tex]\( -x+2 \)[/tex] is also 1.
For rational functions where the degrees of the numerator and the denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients.
The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is -1:
[tex]\[ \text{Horizontal Asymptote:} \ y = \frac{2}{-1} = -2 \][/tex]
So, there is a horizontal asymptote at [tex]\( y = -2 \)[/tex].
### Step 3: Calculate Points on the Function
We need to determine the coordinates of some points on the graph to help plot it. We will substitute specific [tex]\( x \)[/tex]-values into the function to find their corresponding [tex]\( y \)[/tex]-values.
Let's use the points [tex]\( x = -1, 0, 1, \)[/tex] and [tex]\( 3 \)[/tex]:
1. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{2(-1) - 7}{-(-1) + 2} = \frac{-2 - 7}{1 + 2} = \frac{-9}{3} = -3 \][/tex]
Point: [tex]\((-1, -3)\)[/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{2(0) - 7}{-(0) + 2} = \frac{0 - 7}{2} = \frac{-7}{2} = -\frac{7}{2} \][/tex]
Point: [tex]\((0, -\frac{7}{2})\)[/tex]
3. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \frac{2(1) - 7}{-(1) + 2} = \frac{2 - 7}{1} = -5 \][/tex]
Point: [tex]\((1, -5)\)[/tex]
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{2(3) - 7}{-(3) + 2} = \frac{6 - 7}{-1} = \frac{-1}{-1} = 1 \][/tex]
Point: [tex]\((3, 1)\)[/tex]
### Step 4: Plot the Graph
Now that we have all the information, we can sketch the graph:
1. Draw the vertical asymptote [tex]\( x = 2 \)[/tex] as a dashed line.
2. Draw the horizontal asymptote [tex]\( y = -2 \)[/tex] as a dashed line.
3. Plot the points [tex]\((-1, -3)\)[/tex], [tex]\((0, -\frac{7}{2})\)[/tex], [tex]\((1, -5)\)[/tex], and [tex]\((3, 1)\)[/tex].
4. Sketch the curve of [tex]\( f(x) \)[/tex] such that it approaches but never crosses the asymptotes.
Below, I illustrate these points:
```
y
^
| .
| /
| /
| -3 ---> point (-1, -3)
|
-4|.... -----> point (0, -7/2)
| /
|
-2|....--------------------------------- (horizontal asymptote, y = -2)
|
- |
- | /
| .
- | /
| ----/-----> point (1, -5)
-| /
| -----> point (3, 1)
| /
| _________... (vertical asymptote, x = 2)
x
```
This sketch gives an approximate idea of how the function behaves around the asymptotes.
### Step 1: Find the Vertical Asymptote
Vertical asymptotes occur where the denominator of the function is zero and the numerator is not zero. This is because division by zero is undefined.
Given the function:
[tex]\[ f(x) = \frac{2x - 7}{-x + 2} \][/tex]
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 2 = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
### Step 2: Find the Horizontal Asymptote
Horizontal asymptotes describe the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex] or [tex]\( -\infty \)[/tex].
We examine the degrees of the numerator and the denominator:
- The degree of the numerator [tex]\( 2x-7 \)[/tex] is 1.
- The degree of the denominator [tex]\( -x+2 \)[/tex] is also 1.
For rational functions where the degrees of the numerator and the denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients.
The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is -1:
[tex]\[ \text{Horizontal Asymptote:} \ y = \frac{2}{-1} = -2 \][/tex]
So, there is a horizontal asymptote at [tex]\( y = -2 \)[/tex].
### Step 3: Calculate Points on the Function
We need to determine the coordinates of some points on the graph to help plot it. We will substitute specific [tex]\( x \)[/tex]-values into the function to find their corresponding [tex]\( y \)[/tex]-values.
Let's use the points [tex]\( x = -1, 0, 1, \)[/tex] and [tex]\( 3 \)[/tex]:
1. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{2(-1) - 7}{-(-1) + 2} = \frac{-2 - 7}{1 + 2} = \frac{-9}{3} = -3 \][/tex]
Point: [tex]\((-1, -3)\)[/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{2(0) - 7}{-(0) + 2} = \frac{0 - 7}{2} = \frac{-7}{2} = -\frac{7}{2} \][/tex]
Point: [tex]\((0, -\frac{7}{2})\)[/tex]
3. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \frac{2(1) - 7}{-(1) + 2} = \frac{2 - 7}{1} = -5 \][/tex]
Point: [tex]\((1, -5)\)[/tex]
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{2(3) - 7}{-(3) + 2} = \frac{6 - 7}{-1} = \frac{-1}{-1} = 1 \][/tex]
Point: [tex]\((3, 1)\)[/tex]
### Step 4: Plot the Graph
Now that we have all the information, we can sketch the graph:
1. Draw the vertical asymptote [tex]\( x = 2 \)[/tex] as a dashed line.
2. Draw the horizontal asymptote [tex]\( y = -2 \)[/tex] as a dashed line.
3. Plot the points [tex]\((-1, -3)\)[/tex], [tex]\((0, -\frac{7}{2})\)[/tex], [tex]\((1, -5)\)[/tex], and [tex]\((3, 1)\)[/tex].
4. Sketch the curve of [tex]\( f(x) \)[/tex] such that it approaches but never crosses the asymptotes.
Below, I illustrate these points:
```
y
^
| .
| /
| /
| -3 ---> point (-1, -3)
|
-4|.... -----> point (0, -7/2)
| /
|
-2|....--------------------------------- (horizontal asymptote, y = -2)
|
- |
- | /
| .
- | /
| ----/-----> point (1, -5)
-| /
| -----> point (3, 1)
| /
| _________... (vertical asymptote, x = 2)
x
```
This sketch gives an approximate idea of how the function behaves around the asymptotes.
