To identify the correct function based on a graph, we typically look for key characteristics such as amplitude, period, and the type of function (sine or cosine).
- Amplitude: The maximum height of the wave; for sine and cosine functions, this is often 1 unless otherwise scaled.
- Period: This is determined by the coefficient of \(x\) inside the sine or cosine function. The period of \( \sin(kx) \) or \( \cos(kx) \) is given by \( \frac{2\pi}{|k|} \).
Let's analyze the provided options based on these characteristics:
- \( f(x) = \sin(2x) \): Period = \( \frac{2\pi}{2} = \pi \)
- \( f(x) = \cos(4x) \): Period = \( \frac{2\pi}{4} = \frac{\pi}{2} \)
- \( f(x) = \sin(4x) \): Period = \( \frac{2\pi}{4} = \frac{\pi}{2} \)
- \( f(x) = \sin(-4x) \): This is equivalent to \( f(x) = -\sin(4x) \) which reflects the sine function, and the period is also \( \frac{\pi}{2} \).
To determine which function matches a given graph, you would need to:
- Check the amplitude to see if it is 1.
- Observe the period by measuring the distance between repeating wave peaks or troughs.
- Note whether the graph is sine or cosine (sine starts at the origin, cosine starts at its maximum).
If you have a specific graph in mind, please describe it, and I can help you identify which of these functions matches it.