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Sagot :
To analyze and plot the intercepts and asymptotes of the function [tex]\( f(x) = -4 - \frac{1}{x-3} \)[/tex], we follow these steps:
### Step 1: Find the x-intercept
Set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -4 - \frac{1}{x-3} \][/tex]
[tex]\[ 4 = -\frac{1}{x-3} \][/tex]
[tex]\[ -4 = \frac{1}{x-3} \][/tex]
Multiply both sides by [tex]\( x-3 \)[/tex]:
[tex]\[ -4(x-3) = 1 \][/tex]
[tex]\[ -4x + 12 = 1 \][/tex]
[tex]\[ -4x = -11 \][/tex]
[tex]\[ x = \frac{11}{4} \][/tex]
The x-intercept is [tex]\( x = \frac{11}{4} \)[/tex] or [tex]\( 2.75 \)[/tex].
### Step 2: Find the y-intercept
Set [tex]\( x = 0 \)[/tex] and solve for [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = -4 - \frac{1}{0-3} \][/tex]
[tex]\[ f(0) = -4 - \frac{1}{-3} \][/tex]
[tex]\[ f(0) = -4 + \frac{1}{3} \][/tex]
[tex]\[ f(0) = -4 + 0.333 \approx -3.666 \][/tex]
The y-intercept is [tex]\( y \approx -3.666 \)[/tex].
### Step 3: Find the vertical asymptote
A vertical asymptote occurs where the denominator equals zero:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, there is a vertical asymptote at [tex]\( x = 3 \)[/tex].
### Step 4: Plot the function
We'll plot the function along with its intercepts and asymptote:
1. X-Intercept: Plot the point [tex]\((2.75, 0)\)[/tex].
2. Y-Intercept: Plot the point [tex]\((0, -3.666)\)[/tex].
3. Vertical Asymptote: Plot the vertical line [tex]\( x = 3 \)[/tex].
### Summary and Graph:
- The x-intercept is [tex]\( (2.75, 0) \)[/tex]
- The y-intercept is [tex]\( (0, -3.666) \)[/tex]
- The vertical asymptote is at [tex]\( x = 3 \)[/tex]
Here is a conceptual plot:
```
y
^
|
| /
| /
-5 + / x = 3 (vertical asymptote)
| /|
| / |
| / |
| y= -3.666
|
---+---------------- +--------------------> x
0 2.75
```
In this conceptual plot:
- The red star "[tex]\(\ast\)[/tex]" denotes the x-intercept at [tex]\((2.75, 0)\)[/tex].
- The blue star "[tex]\(\ast\)[/tex]" denotes the y-intercept at [tex]\((0, -3.666)\)[/tex].
- The dotted line represents the vertical asymptote at [tex]\(x = 3\)[/tex].
In any plotting tool, substitute these points and the asymptote into the graph to see the function [tex]\( f(x) = -4 - \frac{1}{x-3} \)[/tex].
### Step 1: Find the x-intercept
Set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -4 - \frac{1}{x-3} \][/tex]
[tex]\[ 4 = -\frac{1}{x-3} \][/tex]
[tex]\[ -4 = \frac{1}{x-3} \][/tex]
Multiply both sides by [tex]\( x-3 \)[/tex]:
[tex]\[ -4(x-3) = 1 \][/tex]
[tex]\[ -4x + 12 = 1 \][/tex]
[tex]\[ -4x = -11 \][/tex]
[tex]\[ x = \frac{11}{4} \][/tex]
The x-intercept is [tex]\( x = \frac{11}{4} \)[/tex] or [tex]\( 2.75 \)[/tex].
### Step 2: Find the y-intercept
Set [tex]\( x = 0 \)[/tex] and solve for [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = -4 - \frac{1}{0-3} \][/tex]
[tex]\[ f(0) = -4 - \frac{1}{-3} \][/tex]
[tex]\[ f(0) = -4 + \frac{1}{3} \][/tex]
[tex]\[ f(0) = -4 + 0.333 \approx -3.666 \][/tex]
The y-intercept is [tex]\( y \approx -3.666 \)[/tex].
### Step 3: Find the vertical asymptote
A vertical asymptote occurs where the denominator equals zero:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, there is a vertical asymptote at [tex]\( x = 3 \)[/tex].
### Step 4: Plot the function
We'll plot the function along with its intercepts and asymptote:
1. X-Intercept: Plot the point [tex]\((2.75, 0)\)[/tex].
2. Y-Intercept: Plot the point [tex]\((0, -3.666)\)[/tex].
3. Vertical Asymptote: Plot the vertical line [tex]\( x = 3 \)[/tex].
### Summary and Graph:
- The x-intercept is [tex]\( (2.75, 0) \)[/tex]
- The y-intercept is [tex]\( (0, -3.666) \)[/tex]
- The vertical asymptote is at [tex]\( x = 3 \)[/tex]
Here is a conceptual plot:
```
y
^
|
| /
| /
-5 + / x = 3 (vertical asymptote)
| /|
| / |
| / |
| y= -3.666
|
---+---------------- +--------------------> x
0 2.75
```
In this conceptual plot:
- The red star "[tex]\(\ast\)[/tex]" denotes the x-intercept at [tex]\((2.75, 0)\)[/tex].
- The blue star "[tex]\(\ast\)[/tex]" denotes the y-intercept at [tex]\((0, -3.666)\)[/tex].
- The dotted line represents the vertical asymptote at [tex]\(x = 3\)[/tex].
In any plotting tool, substitute these points and the asymptote into the graph to see the function [tex]\( f(x) = -4 - \frac{1}{x-3} \)[/tex].
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