Answer :
To make a neat sketch of the graph for [tex]\( f(x) = 3\sin(x) \)[/tex] for [tex]\( 0^\circ < x < 360^\circ \)[/tex], follow these steps:
### Step-by-Step Solution
1. Understand the Function:
- The function [tex]\( f(x) = 3\sin(x) \)[/tex] is a scaled sine function.
- The sine function [tex]\( \sin(x) \)[/tex] oscillates between -1 and 1.
- By multiplying by 3, the amplitude of the function changes to oscillate between -3 and 3.
2. Identify Key Points:
- The sine function has specific points where the value is known and these are key to sketching the graph:
- [tex]\( \sin(0^\circ) = 0 \)[/tex]
- [tex]\( \sin(90^\circ) = 1 \)[/tex]
- [tex]\( \sin(180^\circ) = 0 \)[/tex]
- [tex]\( \sin(270^\circ) = -1 \)[/tex]
- [tex]\( \sin(360^\circ) = 0 \)[/tex]
3. Scaling Key Points:
- Multiply the sine values by 3 to fit our function:
- [tex]\( f(0^\circ) = 3\sin(0^\circ) = 0 \)[/tex]
- [tex]\( f(90^\circ) = 3\sin(90^\circ) = 3 \)[/tex]
- [tex]\( f(180^\circ) = 3\sin(180^\circ) = 0 \)[/tex]
- [tex]\( f(270^\circ) = 3\sin(270^\circ) = -3 \)[/tex]
- [tex]\( f(360^\circ) = 3\sin(360^\circ) = 0 \)[/tex]
4. Plot the Points:
- On graph paper or coordinate axis, plot the points:
- (0, 0)
- (90, 3)
- (180, 0)
- (270, -3)
- (360, 0)
5. Draw the Sine Curve:
- Draw a smooth, continuous curve passing through these points, making sure to follow the typical sinusoidal pattern:
- The curve starts at the origin (0, 0), rises to the maximum point (90, 3), falls back to the axis (180, 0), drops to the minimum point (270, -3), and finally rises back to the axis (360, 0).
6. Indicate Intercepts:
- The x-intercepts are at [tex]\( x = 0^\circ, 180^\circ, \)[/tex] and [tex]\( 360^\circ \)[/tex].
- The y-intercept (since the function starts at the origin) is at [tex]\( y = 0 \)[/tex].
### Sketch
Here's a simple way to think of your sketch:
```
|
3 + /\
| / \
2 + / \
| / \
1 + / \
| / \
0 +-----------------------------
| (0,0) (180,0)
-1 + \
| \
-2 + \
| \
-3 + \ / * Intercepts with the x-axis at [tex]\( x = 0^\circ, 180^\circ, 360^\circ \)[/tex].
| (270,-3)
__|________|________|________|________|____________________|_
0 90 180 270 360 (Degrees)
```
### Summary:
- Amplitude is 3, so the highest point is [tex]\( y = 3 \)[/tex] and the lowest is [tex]\( y = -3 \)[/tex].
- X-intercepts are at [tex]\( x = 0^\circ, 180^\circ, 360^\circ \)[/tex].
- Y-intercept is at [tex]\( y = 0 \)[/tex] (origin).
- The function completes one full cycle from [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex].
### Step-by-Step Solution
1. Understand the Function:
- The function [tex]\( f(x) = 3\sin(x) \)[/tex] is a scaled sine function.
- The sine function [tex]\( \sin(x) \)[/tex] oscillates between -1 and 1.
- By multiplying by 3, the amplitude of the function changes to oscillate between -3 and 3.
2. Identify Key Points:
- The sine function has specific points where the value is known and these are key to sketching the graph:
- [tex]\( \sin(0^\circ) = 0 \)[/tex]
- [tex]\( \sin(90^\circ) = 1 \)[/tex]
- [tex]\( \sin(180^\circ) = 0 \)[/tex]
- [tex]\( \sin(270^\circ) = -1 \)[/tex]
- [tex]\( \sin(360^\circ) = 0 \)[/tex]
3. Scaling Key Points:
- Multiply the sine values by 3 to fit our function:
- [tex]\( f(0^\circ) = 3\sin(0^\circ) = 0 \)[/tex]
- [tex]\( f(90^\circ) = 3\sin(90^\circ) = 3 \)[/tex]
- [tex]\( f(180^\circ) = 3\sin(180^\circ) = 0 \)[/tex]
- [tex]\( f(270^\circ) = 3\sin(270^\circ) = -3 \)[/tex]
- [tex]\( f(360^\circ) = 3\sin(360^\circ) = 0 \)[/tex]
4. Plot the Points:
- On graph paper or coordinate axis, plot the points:
- (0, 0)
- (90, 3)
- (180, 0)
- (270, -3)
- (360, 0)
5. Draw the Sine Curve:
- Draw a smooth, continuous curve passing through these points, making sure to follow the typical sinusoidal pattern:
- The curve starts at the origin (0, 0), rises to the maximum point (90, 3), falls back to the axis (180, 0), drops to the minimum point (270, -3), and finally rises back to the axis (360, 0).
6. Indicate Intercepts:
- The x-intercepts are at [tex]\( x = 0^\circ, 180^\circ, \)[/tex] and [tex]\( 360^\circ \)[/tex].
- The y-intercept (since the function starts at the origin) is at [tex]\( y = 0 \)[/tex].
### Sketch
Here's a simple way to think of your sketch:
```
|
3 + /\
| / \
2 + / \
| / \
1 + / \
| / \
0 +-----------------------------
| (0,0) (180,0)
-1 + \
| \
-2 + \
| \
-3 + \ / * Intercepts with the x-axis at [tex]\( x = 0^\circ, 180^\circ, 360^\circ \)[/tex].
| (270,-3)
__|________|________|________|________|____________________|_
0 90 180 270 360 (Degrees)
```
### Summary:
- Amplitude is 3, so the highest point is [tex]\( y = 3 \)[/tex] and the lowest is [tex]\( y = -3 \)[/tex].
- X-intercepts are at [tex]\( x = 0^\circ, 180^\circ, 360^\circ \)[/tex].
- Y-intercept is at [tex]\( y = 0 \)[/tex] (origin).
- The function completes one full cycle from [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex].
