Question

Describe the transformations that happen to the graph of f(x)=sin x to obtain the graph of f(x)=-2sin (x-3)+7. (1 point) vertical stretch of 2, reflect over the x-axis, right 3 units, and up 7 units vertical stretch of 2, reflect over the y-axis, right 3 units, and down 7 units vertical stretch of 2, reflect over the y-axis, left 3 units, and up 7 units vertical stretch of 2, reflect over the y-axis, right 3 units, and up 7 units
Which option is correct
Answer
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Answered by GPT-4o mini
To transform the graph of \( f(x) = \sin x \) into the graph of \( f(x) = -2\sin(x-3) + 7 \), we follow these steps:

1. **Horizontal Shift**: The term \( (x - 3) \) indicates a right shift of 3 units.
2. **Vertical Stretch and Reflection**: The coefficient of -2 means that the sine function is vertically stretched by a factor of 2 and reflected over the x-axis.
3. **Vertical Shift**: The addition of +7 indicates that the entire graph is shifted up by 7 units.

Putting this all together, the transformations can be summarized as:
- Reflect over the x-axis (due to the negative sign).
- Vertical stretch by a factor of 2.
- Right 3 units (due to \( x - 3 \)).
- Up 7 units (due to the +7).

Based on these steps, the correct option is:
**Vertical stretch of 2, reflect over the x-axis, right 3 units, and up 7 units.**
Answered by GPT-4o mini
Yes, let's break down the transformations step by step more clearly to confirm.

Starting with the basic function \( f(x) = \sin x \):

1. **Horizontal Shift**: The presence of \( (x - 3) \) indicates a right shift by 3 units. This means the function \( \sin x \) will now evaluate its sine at \( x - 3 \).

2. **Vertical Stretch and Reflection**: The factor of \(-2\) in front of the sine function indicates two transformations:
- The negative sign reflects the graph over the x-axis.
- The factor of \(2\) indicates a vertical stretch, making the peaks and troughs of the sine wave taller by a factor of 2.

3. **Vertical Shift**: The \(+7\) at the end shifts the entire graph up by 7 units.

So, summarizing those transformations:
- **Right 3 units** (due to \(x - 3\))
- **Reflect over the x-axis** (due to \(-\))
- **Vertical stretch by a factor of 2**
- **Up 7 units** (due to \(+7\))

Therefore, the correct description is:
**Vertical stretch of 2, reflect over the x-axis, right 3 units, and up 7 units.**

Thank you for your patience!