Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Let's sketch the graph of the function [tex]\( f(x) = \sin(x) + 2 \)[/tex] for [tex]\( 0^\circ < x < 360^\circ \)[/tex].
### Step-by-Step Solution:
1. Identify the Function Form:
The given function [tex]\( f(x) = \sin(x) + 2 \)[/tex] represents a sine wave that has been vertically shifted upwards by 2 units.
2. Determine Intercepts and Characteristics:
- Y-intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = \sin(0) + 2 = 0 + 2 = 2 \][/tex]
So, the y-intercept is at [tex]\((0, 2)\)[/tex].
- X-intercepts:
To find the x-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \sin(x) + 2 = 0 \quad \Rightarrow \quad \sin(x) = -2 \][/tex]
Since [tex]\( \sin(x) \)[/tex] ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex], there are no x-intercepts as [tex]\( \sin(x) = -2 \)[/tex] is impossible within the given range.
3. Determine the Maximum and Minimum Points:
- The maximum value of [tex]\( \sin(x) \)[/tex] is [tex]\(1\)[/tex]. Therefore, the maximum value of [tex]\( f(x) \)[/tex] is:
[tex]\[ f_{\text{max}} = 1 + 2 = 3 \][/tex]
This occurs at [tex]\( x = 90^\circ \)[/tex] and [tex]\( x = 270^\circ \)[/tex].
- The minimum value of [tex]\( \sin(x) \)[/tex] is [tex]\(-1\)[/tex]. Therefore, the minimum value of [tex]\( f(x) \)[/tex] is:
[tex]\[ f_{\text{min}} = -1 + 2 = 1 \][/tex]
This occurs at [tex]\( x = 180^\circ \)[/tex] and [tex]\( x = 360^\circ \)[/tex].
4. Plot the Function Values:
- At [tex]\( x = 0^\circ \)[/tex], [tex]\( f(0) = 2 \)[/tex]
- At [tex]\( x = 90^\circ \)[/tex], [tex]\( f(90) = 3 \)[/tex]
- At [tex]\( x = 180^\circ \)[/tex], [tex]\( f(180) = 1 \)[/tex]
- At [tex]\( x = 270^\circ \)[/tex], [tex]\( f(270) = 3 \)[/tex]
- At [tex]\( x = 360^\circ \)[/tex], [tex]\( f(360) = 2 \)[/tex]
5. Sketch the Graph:
- Draw the x-axis ranging from [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex].
- Draw the y-axis and mark the intercept at [tex]\((0, 2)\)[/tex].
- Plot points at [tex]\((90^\circ, 3)\)[/tex], [tex]\((180^\circ, 1)\)[/tex], [tex]\((270^\circ, 3)\)[/tex], and [tex]\((360^\circ, 2)\)[/tex].
- Sketch a smooth curve through these points to illustrate the sine wave shifted up by 2 units.
### Sketch of [tex]\( f(x) = \sin(x) + 2 \)[/tex]:
```
y
^
3 |
|
|
2 |------------------------------------------------> x
|
1 | *
|
|
0 +------------------------------------------------------
0° 90° 180° 270° 360°
```
In the graph:
- The y-intercept is clearly marked at [tex]\((0, 2)\)[/tex].
- The curve shows one full period of the sine function shifted up by 2 units, peaking at [tex]\(90^\circ\)[/tex] and [tex]\(270^\circ\)[/tex].
The sketch accurately represents [tex]\( f(x) = \sin(x) + 2 \)[/tex] over the interval [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex].
### Step-by-Step Solution:
1. Identify the Function Form:
The given function [tex]\( f(x) = \sin(x) + 2 \)[/tex] represents a sine wave that has been vertically shifted upwards by 2 units.
2. Determine Intercepts and Characteristics:
- Y-intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = \sin(0) + 2 = 0 + 2 = 2 \][/tex]
So, the y-intercept is at [tex]\((0, 2)\)[/tex].
- X-intercepts:
To find the x-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \sin(x) + 2 = 0 \quad \Rightarrow \quad \sin(x) = -2 \][/tex]
Since [tex]\( \sin(x) \)[/tex] ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex], there are no x-intercepts as [tex]\( \sin(x) = -2 \)[/tex] is impossible within the given range.
3. Determine the Maximum and Minimum Points:
- The maximum value of [tex]\( \sin(x) \)[/tex] is [tex]\(1\)[/tex]. Therefore, the maximum value of [tex]\( f(x) \)[/tex] is:
[tex]\[ f_{\text{max}} = 1 + 2 = 3 \][/tex]
This occurs at [tex]\( x = 90^\circ \)[/tex] and [tex]\( x = 270^\circ \)[/tex].
- The minimum value of [tex]\( \sin(x) \)[/tex] is [tex]\(-1\)[/tex]. Therefore, the minimum value of [tex]\( f(x) \)[/tex] is:
[tex]\[ f_{\text{min}} = -1 + 2 = 1 \][/tex]
This occurs at [tex]\( x = 180^\circ \)[/tex] and [tex]\( x = 360^\circ \)[/tex].
4. Plot the Function Values:
- At [tex]\( x = 0^\circ \)[/tex], [tex]\( f(0) = 2 \)[/tex]
- At [tex]\( x = 90^\circ \)[/tex], [tex]\( f(90) = 3 \)[/tex]
- At [tex]\( x = 180^\circ \)[/tex], [tex]\( f(180) = 1 \)[/tex]
- At [tex]\( x = 270^\circ \)[/tex], [tex]\( f(270) = 3 \)[/tex]
- At [tex]\( x = 360^\circ \)[/tex], [tex]\( f(360) = 2 \)[/tex]
5. Sketch the Graph:
- Draw the x-axis ranging from [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex].
- Draw the y-axis and mark the intercept at [tex]\((0, 2)\)[/tex].
- Plot points at [tex]\((90^\circ, 3)\)[/tex], [tex]\((180^\circ, 1)\)[/tex], [tex]\((270^\circ, 3)\)[/tex], and [tex]\((360^\circ, 2)\)[/tex].
- Sketch a smooth curve through these points to illustrate the sine wave shifted up by 2 units.
### Sketch of [tex]\( f(x) = \sin(x) + 2 \)[/tex]:
```
y
^
3 |
|
|
2 |------------------------------------------------> x
|
1 | *
|
|
0 +------------------------------------------------------
0° 90° 180° 270° 360°
```
In the graph:
- The y-intercept is clearly marked at [tex]\((0, 2)\)[/tex].
- The curve shows one full period of the sine function shifted up by 2 units, peaking at [tex]\(90^\circ\)[/tex] and [tex]\(270^\circ\)[/tex].
The sketch accurately represents [tex]\( f(x) = \sin(x) + 2 \)[/tex] over the interval [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex].
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
