Which function is graphed below?
On a coordinate plane, a curve goes through (0, 0), has a maximum point at 1, and has a minimum point at -1. It increases through (0, 0). A. f(x)=−cos(x) B. f(x)=cos(x) C. f(x)=sin(x) D. f(x)=−sin(x)
The Correct Answer and Explanation is:
The function that best describes the curve graphed in the description is D. f(x) = -sin(x).
To understand why this is the correct choice, we need to analyze the properties of the sine function and how the negative transformation affects its graph. The function (f(x) = \sin(x)) oscillates between -1 and 1, with the following characteristics:
- Behavior of the Sine Function: The standard sine function, (f(x) = \sin(x)), starts at (0, 0) and rises to a maximum of 1 at (x = \frac{\pi}{2}) before descending back to 0 at (x = \pi). It has a minimum point of -1 at (x = \frac{3\pi}{2}).
- Effect of Negation: When we consider the function (f(x) = -\sin(x)), we essentially reflect the sine curve across the x-axis. This transformation alters the behavior significantly: the function now starts at (0, 0) and initially decreases to a minimum of -1 at (x = \frac{\pi}{2}) before returning to 0 at (x = \pi), and then it increases to a maximum of 1 at (x = \frac{3\pi}{2}).
- Maximum and Minimum Points: Given the description in the problem, the function passes through (0, 0), achieves a maximum point at (1, 0), and a minimum at (-1, 0). The characteristics of the graph described—having a maximum at (y = 1) and a minimum at (y = -1)—fit perfectly with the behavior of (f(x) = -\sin(x)).
In conclusion, based on the given features—starting point, maximum and minimum points, and the general behavior of the sine function under negation—the graph corresponds to the function D. f(x) = -sin(x).