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Sagot :
To plot the intercepts and asymptotes of the rational function [tex]\( f(x) = \frac{5}{x-4} - 4 \)[/tex], let's go through the steps one by one.
### 1. Vertical Asymptote
The vertical asymptote occurs where the denominator of the rational function equals zero because at this point, the function tends toward infinity.
For [tex]\( f(x) = \frac{5}{x - 4} - 4 \)[/tex], set the denominator to zero:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, the vertical asymptote is at [tex]\( x = 4 \)[/tex].
### 2. Horizontal Asymptote
The horizontal asymptote of a rational function [tex]\( \frac{P(x)}{Q(x)} \)[/tex] can be found by comparing the degrees of [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex]. Here, [tex]\( P(x) \)[/tex] is the numerator [tex]\( 5 \)[/tex], and [tex]\( Q(x) \)[/tex] is the denominator [tex]\( x - 4 \)[/tex].
For the given function:
[tex]\[ f(x) = \frac{5}{x - 4} - 4 \][/tex]
As [tex]\( x \)[/tex] tends to positive or negative infinity, the term [tex]\( \frac{5}{x - 4} \)[/tex] approaches zero. Therefore, the function approaches:
[tex]\[ f(x) \approx -4 \][/tex]
So, the horizontal asymptote is [tex]\( y = -4 \)[/tex].
### 3. [tex]\( x \)[/tex]-Intercept
The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex]. Set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{5}{x - 4} - 4 = 0 \][/tex]
First, isolate the fraction:
[tex]\[ \frac{5}{x - 4} = 4 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ 5 = 4(x - 4) \][/tex]
[tex]\[ 5 = 4x - 16 \][/tex]
[tex]\[ 4x = 21 \][/tex]
[tex]\[ x = \frac{21}{4} \][/tex]
So, the [tex]\( x \)[/tex]-intercept is [tex]\( x = \frac{21}{4} \)[/tex] or approximately [tex]\( x = 5.25 \)[/tex].
### 4. [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. Evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{5}{0 - 4} - 4 \][/tex]
[tex]\[ f(0) = \frac{5}{-4} - 4 \][/tex]
[tex]\[ f(0) = -\frac{5}{4} - 4 \][/tex]
[tex]\[ f(0) = -\frac{5}{4} - \frac{16}{4} \][/tex]
[tex]\[ f(0) = -\frac{21}{4} \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( y = -\frac{21}{4} \)[/tex] or approximately [tex]\( y = -5.25 \)[/tex].
### Summary of Intercepts and Asymptotes
- Vertical Asymptote: [tex]\( x = 4 \)[/tex]
- Horizontal Asymptote: [tex]\( y = -4 \)[/tex]
- [tex]\( x \)[/tex]-Intercept: [tex]\( x = \frac{21}{4} \)[/tex] or approximately [tex]\( x = 5.25 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( y = -\frac{21}{4} \)[/tex] or approximately [tex]\( y = -5.25 \)[/tex]
### Plotting:
1. Draw the vertical asymptote as a dashed line at [tex]\( x = 4 \)[/tex].
2. Draw the horizontal asymptote as a dashed line at [tex]\( y = -4 \)[/tex].
3. Mark the [tex]\( x \)[/tex]-intercept at [tex]\( (5.25, 0) \)[/tex].
4. Mark the [tex]\( y \)[/tex]-intercept at [tex]\( (0, -5.25) \)[/tex].
5. Sketch the curve approaching these asymptotes while passing through the intercepts.
Here is a rough sketch of the function based on the given data:
```plaintext
63 | .
|
|
|
|
|
|
|
|
|
21 |-------------------------------------------------------
|
|
| .
| \
| ....
|................................ ..........
| ....
| .
|
-21 |_______________________________________________________
-10 0 10
```
This plot shows the asymptotes, intercepts, and the general behavior of [tex]\( f(x) = \frac{5}{x - 4} - 4 \)[/tex].
### 1. Vertical Asymptote
The vertical asymptote occurs where the denominator of the rational function equals zero because at this point, the function tends toward infinity.
For [tex]\( f(x) = \frac{5}{x - 4} - 4 \)[/tex], set the denominator to zero:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, the vertical asymptote is at [tex]\( x = 4 \)[/tex].
### 2. Horizontal Asymptote
The horizontal asymptote of a rational function [tex]\( \frac{P(x)}{Q(x)} \)[/tex] can be found by comparing the degrees of [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex]. Here, [tex]\( P(x) \)[/tex] is the numerator [tex]\( 5 \)[/tex], and [tex]\( Q(x) \)[/tex] is the denominator [tex]\( x - 4 \)[/tex].
For the given function:
[tex]\[ f(x) = \frac{5}{x - 4} - 4 \][/tex]
As [tex]\( x \)[/tex] tends to positive or negative infinity, the term [tex]\( \frac{5}{x - 4} \)[/tex] approaches zero. Therefore, the function approaches:
[tex]\[ f(x) \approx -4 \][/tex]
So, the horizontal asymptote is [tex]\( y = -4 \)[/tex].
### 3. [tex]\( x \)[/tex]-Intercept
The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex]. Set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{5}{x - 4} - 4 = 0 \][/tex]
First, isolate the fraction:
[tex]\[ \frac{5}{x - 4} = 4 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ 5 = 4(x - 4) \][/tex]
[tex]\[ 5 = 4x - 16 \][/tex]
[tex]\[ 4x = 21 \][/tex]
[tex]\[ x = \frac{21}{4} \][/tex]
So, the [tex]\( x \)[/tex]-intercept is [tex]\( x = \frac{21}{4} \)[/tex] or approximately [tex]\( x = 5.25 \)[/tex].
### 4. [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. Evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{5}{0 - 4} - 4 \][/tex]
[tex]\[ f(0) = \frac{5}{-4} - 4 \][/tex]
[tex]\[ f(0) = -\frac{5}{4} - 4 \][/tex]
[tex]\[ f(0) = -\frac{5}{4} - \frac{16}{4} \][/tex]
[tex]\[ f(0) = -\frac{21}{4} \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( y = -\frac{21}{4} \)[/tex] or approximately [tex]\( y = -5.25 \)[/tex].
### Summary of Intercepts and Asymptotes
- Vertical Asymptote: [tex]\( x = 4 \)[/tex]
- Horizontal Asymptote: [tex]\( y = -4 \)[/tex]
- [tex]\( x \)[/tex]-Intercept: [tex]\( x = \frac{21}{4} \)[/tex] or approximately [tex]\( x = 5.25 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( y = -\frac{21}{4} \)[/tex] or approximately [tex]\( y = -5.25 \)[/tex]
### Plotting:
1. Draw the vertical asymptote as a dashed line at [tex]\( x = 4 \)[/tex].
2. Draw the horizontal asymptote as a dashed line at [tex]\( y = -4 \)[/tex].
3. Mark the [tex]\( x \)[/tex]-intercept at [tex]\( (5.25, 0) \)[/tex].
4. Mark the [tex]\( y \)[/tex]-intercept at [tex]\( (0, -5.25) \)[/tex].
5. Sketch the curve approaching these asymptotes while passing through the intercepts.
Here is a rough sketch of the function based on the given data:
```plaintext
63 | .
|
|
|
|
|
|
|
|
|
21 |-------------------------------------------------------
|
|
| .
| \
| ....
|................................ ..........
| ....
| .
|
-21 |_______________________________________________________
-10 0 10
```
This plot shows the asymptotes, intercepts, and the general behavior of [tex]\( f(x) = \frac{5}{x - 4} - 4 \)[/tex].
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