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Sagot :
To sketch the graph of the function [tex]\( f(x) = -3 \tan x \)[/tex] for [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex], let's break down the steps necessary to understand and plot this function.
### Understanding the Tangent Function
1. Basics of [tex]\( \tan x \)[/tex]:
- The tangent function [tex]\( \tan x \)[/tex] has a period of [tex]\( 180^\circ \)[/tex].
- It is undefined at [tex]\( x = 90^\circ + n \cdot 180^\circ \)[/tex] for any integer [tex]\( n \)[/tex] because the cosine function in its denominator is zero at those points.
2. Properties of [tex]\( -\tan x \)[/tex]:
- Multiplying by -1 reflects the graph of [tex]\( \tan x \)[/tex] about the x-axis.
3. Scaling by -3:
- Multiplying by -3 scales the graph vertically, stretching it by a factor of 3 and reflecting it about the x-axis.
### Key Points and Intercepts
To sketch [tex]\( f(x) = -3 \tan x \)[/tex]:
#### Intercepts
1. x-intercepts:
- The function [tex]\( \tan x \)[/tex] is 0 at [tex]\( x = 0^\circ, 180^\circ, 360^\circ \)[/tex].
- Therefore, [tex]\( -3 \tan x \)[/tex] is also 0 at these points.
- Thus, the x-intercepts are at [tex]\( (0^\circ, 0), (180^\circ, 0), (360^\circ, 0) \)[/tex].
2. y-intercepts:
- At [tex]\( x = 0^\circ \)[/tex], [tex]\( f(x) = -3 \tan 0 = 0 \)[/tex].
- So the y-intercept is at [tex]\( (0, 0) \)[/tex].
#### Asymptotes
- Vertical Asymptotes:
- The function has vertical asymptotes where [tex]\( \tan x \)[/tex] is undefined.
- These points are at [tex]\( 90^\circ + n \cdot 180^\circ \)[/tex].
- For the given range [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex], the asymptotes are at [tex]\( x = 90^\circ, 270^\circ \)[/tex].
### Plotting Points between Asymptotes
- Behavior Between Asymptotes:
- Between [tex]\( 0^\circ \)[/tex] and [tex]\( 90^\circ \)[/tex], [tex]\( \tan x \)[/tex] increases from 0 to [tex]\( \infty \)[/tex], thus [tex]\( -3 \tan x \)[/tex] decreases from 0 to [tex]\( -\infty \)[/tex].
- Between [tex]\( 90^\circ \)[/tex] and [tex]\( 180^\circ \)[/tex], [tex]\( \tan x \)[/tex] decreases from [tex]\( -\infty \)[/tex] to 0, thus [tex]\( -3 \tan x \)[/tex] increases from [tex]\( \infty \)[/tex] to 0.
- This repeating pattern occurs for each period of [tex]\( \tan x \)[/tex].
### Steps to Sketch the Graph
1. Draw the x-axis from [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex] and mark the points where vertical asymptotes occur at [tex]\( 90^\circ \)[/tex] and [tex]\( 270^\circ \)[/tex].
2. Mark the x-intercepts at [tex]\( (0^\circ, 0), (180^\circ, 0), (360^\circ, 0) \)[/tex].
3. Sketch the curve:
- From 0 to 90 degrees, the curve starts at the origin and goes down to [tex]\( -\infty \)[/tex].
- From 90 to 180 degrees, start from [tex]\( +\infty \)[/tex], decrease to 0 at 180 degrees.
- From 180 to 270 degrees, the curve starts at 0, goes down to [tex]\( -\infty \)[/tex].
- From 270 to 360 degrees, start from [tex]\( +\infty \)[/tex], finish at 0.
Here's a rough visualization:
```
∞ | | | ∞
-3 | | |
| | |
------|----------|----------|--------
-3 | | |
| | |
------------|----------|----------|---
| | |
( 0,0) (90,∞) (180,0) (270,∞) (360,0)
```
### Concluding with Asymptotes and Intercepts
Now that we've plotted this, we:
- Clearly show all intercepts: (0, 0), (180, 0), (360, 0).
- Clearly show vertical asymptotes: [tex]\( x = 90^\circ \)[/tex], [tex]\( x = 270^\circ \)[/tex].
- Ensure both axes and important points are labeled, and the behavior around the asymptotes is properly indicated.
This gives a full sketch of [tex]\( f(x) = -3 \tan x \)[/tex] in the interval [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex].
### Understanding the Tangent Function
1. Basics of [tex]\( \tan x \)[/tex]:
- The tangent function [tex]\( \tan x \)[/tex] has a period of [tex]\( 180^\circ \)[/tex].
- It is undefined at [tex]\( x = 90^\circ + n \cdot 180^\circ \)[/tex] for any integer [tex]\( n \)[/tex] because the cosine function in its denominator is zero at those points.
2. Properties of [tex]\( -\tan x \)[/tex]:
- Multiplying by -1 reflects the graph of [tex]\( \tan x \)[/tex] about the x-axis.
3. Scaling by -3:
- Multiplying by -3 scales the graph vertically, stretching it by a factor of 3 and reflecting it about the x-axis.
### Key Points and Intercepts
To sketch [tex]\( f(x) = -3 \tan x \)[/tex]:
#### Intercepts
1. x-intercepts:
- The function [tex]\( \tan x \)[/tex] is 0 at [tex]\( x = 0^\circ, 180^\circ, 360^\circ \)[/tex].
- Therefore, [tex]\( -3 \tan x \)[/tex] is also 0 at these points.
- Thus, the x-intercepts are at [tex]\( (0^\circ, 0), (180^\circ, 0), (360^\circ, 0) \)[/tex].
2. y-intercepts:
- At [tex]\( x = 0^\circ \)[/tex], [tex]\( f(x) = -3 \tan 0 = 0 \)[/tex].
- So the y-intercept is at [tex]\( (0, 0) \)[/tex].
#### Asymptotes
- Vertical Asymptotes:
- The function has vertical asymptotes where [tex]\( \tan x \)[/tex] is undefined.
- These points are at [tex]\( 90^\circ + n \cdot 180^\circ \)[/tex].
- For the given range [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex], the asymptotes are at [tex]\( x = 90^\circ, 270^\circ \)[/tex].
### Plotting Points between Asymptotes
- Behavior Between Asymptotes:
- Between [tex]\( 0^\circ \)[/tex] and [tex]\( 90^\circ \)[/tex], [tex]\( \tan x \)[/tex] increases from 0 to [tex]\( \infty \)[/tex], thus [tex]\( -3 \tan x \)[/tex] decreases from 0 to [tex]\( -\infty \)[/tex].
- Between [tex]\( 90^\circ \)[/tex] and [tex]\( 180^\circ \)[/tex], [tex]\( \tan x \)[/tex] decreases from [tex]\( -\infty \)[/tex] to 0, thus [tex]\( -3 \tan x \)[/tex] increases from [tex]\( \infty \)[/tex] to 0.
- This repeating pattern occurs for each period of [tex]\( \tan x \)[/tex].
### Steps to Sketch the Graph
1. Draw the x-axis from [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex] and mark the points where vertical asymptotes occur at [tex]\( 90^\circ \)[/tex] and [tex]\( 270^\circ \)[/tex].
2. Mark the x-intercepts at [tex]\( (0^\circ, 0), (180^\circ, 0), (360^\circ, 0) \)[/tex].
3. Sketch the curve:
- From 0 to 90 degrees, the curve starts at the origin and goes down to [tex]\( -\infty \)[/tex].
- From 90 to 180 degrees, start from [tex]\( +\infty \)[/tex], decrease to 0 at 180 degrees.
- From 180 to 270 degrees, the curve starts at 0, goes down to [tex]\( -\infty \)[/tex].
- From 270 to 360 degrees, start from [tex]\( +\infty \)[/tex], finish at 0.
Here's a rough visualization:
```
∞ | | | ∞
-3 | | |
| | |
------|----------|----------|--------
-3 | | |
| | |
------------|----------|----------|---
| | |
( 0,0) (90,∞) (180,0) (270,∞) (360,0)
```
### Concluding with Asymptotes and Intercepts
Now that we've plotted this, we:
- Clearly show all intercepts: (0, 0), (180, 0), (360, 0).
- Clearly show vertical asymptotes: [tex]\( x = 90^\circ \)[/tex], [tex]\( x = 270^\circ \)[/tex].
- Ensure both axes and important points are labeled, and the behavior around the asymptotes is properly indicated.
This gives a full sketch of [tex]\( f(x) = -3 \tan x \)[/tex] in the interval [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex].
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