Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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Can you show me step by step how to solve the problem and sketch the graph
Transcribed Image Text:### Problem Description:
3. Sketch the graph of \( f(x) = 2 \sin(x) + 3 \)
### Explanation:
In this problem, we are asked to sketch the graph of the function \( f(x) = 2 \sin(x) + 3 \). Here's how to approach graphing this function:
1. **Identify the base function**: The base function here is \( \sin(x) \), which is a standard sine function.
2. **Amplitude Adjustment**: The coefficient 2 in front of \( \sin(x) \) affects the amplitude of the sine wave. Instead of oscillating between -1 and 1, the sine wave will now oscillate between -2 and 2.
3. **Vertical Shift**: The addition of 3 shifts the entire graph vertically upward by 3 units. Hence, instead of oscillating around the x-axis, the graph will now oscillate around the line \( y = 3 \).
4. **Period and Frequency**: Since there is no coefficient affecting the \( x \) inside the sine function, the period and frequency remain unchanged. The standard period of \( \sin(x) \) is \( 2\pi \).
### Graph Description:
To graph \( f(x) = 2 \sin(x) + 3 \):
- Start with the basic shape of \( \sin(x) \), which oscillates between -1 and 1.
- Scale the amplitude so that the wave oscillates between -2 and 2.
- Shift the entire graph up by 3 units so the centerline of the wave is at \( y = 3 \).
- Mark key points such as the maximum (5), minimum (1), and intercept at each half-period (when \( \sin(x) \) is 0, \( f(x) = 3 \)).
The resulting graph will be a sine wave oscillating between 1 and 5 with a central axis at \( y = 3 \).
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