Question
Ms. Arrowood and Mrs. Owen both described the graph of the function y = sin x − 3 .
ms arrowood says the graph is the sin graph shifted right 3 units . and Mrs owen I think the sin graph is shifted down three units, which one is correct ? and how do you know who is correct ?
ms arrowood says the graph is the sin graph shifted right 3 units . and Mrs owen I think the sin graph is shifted down three units, which one is correct ? and how do you know who is correct ?
Answers
GPT-4o mini
To determine which description of the function \( y = \sin x - 3 \) is correct, let’s analyze the function step by step.
1. **Understand the basic sine function**: The function \( y = \sin x \) oscillates between -1 and 1 with a period of \( 2\pi \).
2. **Identify the transformation**: The expression \( y = \sin x - 3 \) indicates a vertical transformation, specifically a vertical shift.
3. **Shifting the sine function**:
- Subtracting \( 3 \) from \( \sin x \) means that every point on the sine graph is shifted down by 3 units.
- Therefore, instead of oscillating between -1 and 1, the new function \( y = \sin x - 3 \) will oscillate between -4 and -2.
So, Mrs. Owen is correct: the graph of the function is the sine graph shifted **down** 3 units.
Ms. Arrowood’s statement about the graph being shifted **right** 3 units is incorrect, as there is no horizontal transformation (right or left shift) indicated in the function \( y = \sin x - 3 \).
In summary, \( y = \sin x - 3 \) represents a vertical shift down by 3 units, confirming Mrs. Owen's statement.
1. **Understand the basic sine function**: The function \( y = \sin x \) oscillates between -1 and 1 with a period of \( 2\pi \).
2. **Identify the transformation**: The expression \( y = \sin x - 3 \) indicates a vertical transformation, specifically a vertical shift.
3. **Shifting the sine function**:
- Subtracting \( 3 \) from \( \sin x \) means that every point on the sine graph is shifted down by 3 units.
- Therefore, instead of oscillating between -1 and 1, the new function \( y = \sin x - 3 \) will oscillate between -4 and -2.
So, Mrs. Owen is correct: the graph of the function is the sine graph shifted **down** 3 units.
Ms. Arrowood’s statement about the graph being shifted **right** 3 units is incorrect, as there is no horizontal transformation (right or left shift) indicated in the function \( y = \sin x - 3 \).
In summary, \( y = \sin x - 3 \) represents a vertical shift down by 3 units, confirming Mrs. Owen's statement.