Question
f(x) = sin x + 2 describe the transformation of the graph f(x)what would be the f(-x)
Answers
GPT-4o mini
To analyze the function \( f(x) = \sin x + 2 \), we can break it down into its transformations:
1. **Basic Function**: The basic function here is \( \sin x \), which has a range of \([-1, 1]\).
2. **Vertical Shift**: The \( +2 \) shifts the entire graph of \( \sin x \) upward by 2 units. As a result, the new range of the function \( f(x) = \sin x + 2 \) becomes \([1, 3]\).
Now, let's analyze \( f(-x) \):
3. **Reflection**: The expression \( f(-x) \) implies a reflection of the graph across the y-axis. This means we replace \( x \) with \( -x \) in the original function:
\[
f(-x) = \sin(-x) + 2
\]
Using the property of the sine function, which is odd (\(\sin(-x) = -\sin(x)\)), we can simplify:
\[
f(-x) = -\sin(x) + 2
\]
### Summary:
- The transformation of the graph for \( f(x) = \sin x + 2 \) includes a vertical shift upward by 2 units.
- The function \( f(-x) = -\sin(x) + 2 \) reflects the graph of \( f(x) \) across the y-axis and also reflects it downward about the horizontal line \( y = 2 \).
Overall, while \( f(x) \) is shifted upward, \( f(-x) \) combines both a vertical reflection and a change in orientation.
1. **Basic Function**: The basic function here is \( \sin x \), which has a range of \([-1, 1]\).
2. **Vertical Shift**: The \( +2 \) shifts the entire graph of \( \sin x \) upward by 2 units. As a result, the new range of the function \( f(x) = \sin x + 2 \) becomes \([1, 3]\).
Now, let's analyze \( f(-x) \):
3. **Reflection**: The expression \( f(-x) \) implies a reflection of the graph across the y-axis. This means we replace \( x \) with \( -x \) in the original function:
\[
f(-x) = \sin(-x) + 2
\]
Using the property of the sine function, which is odd (\(\sin(-x) = -\sin(x)\)), we can simplify:
\[
f(-x) = -\sin(x) + 2
\]
### Summary:
- The transformation of the graph for \( f(x) = \sin x + 2 \) includes a vertical shift upward by 2 units.
- The function \( f(-x) = -\sin(x) + 2 \) reflects the graph of \( f(x) \) across the y-axis and also reflects it downward about the horizontal line \( y = 2 \).
Overall, while \( f(x) \) is shifted upward, \( f(-x) \) combines both a vertical reflection and a change in orientation.