Graph each function over a one-period interval.
y = -4 sin(2x - π)

Verified step by step guidance

1. The standard form of a sine function is \( y = a \sin(bx - c) + d \). In this case, \( a = -4 \), \( b = 2 \), \( c = \pi \), and \( d = 0 \).
2. The amplitude of the function is the absolute value of \( a \), which is \( |a| = |-4| = 4 \).
3. The period of a sine function is given by \( \frac{2\pi}{b} \). Here, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
4. The phase shift is calculated by \( \frac{c}{b} \). With \( c = \pi \) and \( b = 2 \), the phase shift is \( \frac{\pi}{2} \) to the right.
5. Start by plotting the key points of the sine function over one period \([0, \pi]\), considering the amplitude, period, and phase shift. Reflect the graph across the x-axis due to the negative amplitude.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is periodic, with a standard period of 2π, meaning it repeats its values every 2π units. Understanding the sine function's behavior is crucial for graphing transformations and identifying key features such as amplitude and phase shift.

Amplitude and Vertical Shift

Amplitude refers to the maximum distance a wave reaches from its central axis, which in the case of the sine function is determined by the coefficient in front of the sine term. In the function y = -4 sin(2x - π), the amplitude is 4, indicating the graph will oscillate between 4 and -4. The negative sign indicates a reflection over the x-axis, affecting the direction of the wave.

Phase Shift and Frequency

Phase shift refers to the horizontal shift of the graph of a function, determined by the value subtracted from the variable inside the function. In y = -4 sin(2x - π), the term (2x - π) indicates a phase shift of π/2 to the right, as it can be rewritten as sin(2(x - π/2)). The coefficient 2 in front of x also indicates the frequency, which affects how many cycles occur in a given interval.