Let's analyze the function given: [tex]\( g(x) = \sin(-3x) + 2 \)[/tex].
To understand how the sine graph is transformed, follow these detailed steps:
1. Reflection and Stretching/Compression:
- The term [tex]\(-3x\)[/tex] inside the sine function indicates a horizontal transformation.
- The coefficient [tex]\(-3\)[/tex] means two things:
1. The negative sign causes a reflection over the y-axis. For sine, this means [tex]\(\sin(-x) = -\sin(x)\)[/tex].
2. The coefficient 3 indicates a horizontal compression by a factor of 3. This changes the period of the sine function. The period of [tex]\(\sin(x)\)[/tex] is [tex]\(2\pi\)[/tex], but for [tex]\(\sin(3x)\)[/tex], the period becomes [tex]\(\frac{2\pi}{3}\)[/tex].
2. Vertical Shift:
- The [tex]\(+2\)[/tex] outside the sine function represents a vertical shift. It moves the entire graph of the sine function up by 2 units.
Putting all these transformations together:
- Start with the basic sine curve.
- Reflect it over the y-axis (though for the sine function it remains symmetric).
- Compress it horizontally so its new period is [tex]\(\frac{2\pi}{3}\)[/tex]. This means it completes one sine wave every [tex]\(\frac{2\pi}{3}\)[/tex] units on the x-axis.
- Shift the resulting curve upward by 2 units.
The result is a sine wave that oscillates between 1 and 3 (since it would normally oscillate between -1 and 1, the upward shift moves this range to 1 to 3), completes a full cycle every [tex]\(\frac{2\pi}{3}\)[/tex] units, and is reflected across the y-axis.
To find the correct graph, look for a graph that:
- Oscillates between 1 and 3.
- Has a period of [tex]\(\frac{2\pi}{3}\)[/tex].
- Is shifted up by 2 units vertically.
Without seeing the actual graphs A and B, I am unable to select one. However, knowing these transformations, you should be able to pick the graph that matches these criteria.
