In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = -3 sin 2πx

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1. Identify the general form of the sine function: \( y = a \sin(bx + c) + d \). In this case, \( y = -3 \sin(2\pi x) \).
2. Determine the amplitude of the function. The amplitude is the absolute value of the coefficient \( a \). Here, \( a = -3 \), so the amplitude is \( |a| = 3 \).
3. Find the period of the function. The period of a sine function is given by \( \frac{2\pi}{b} \). In this function, \( b = 2\pi \), so the period is \( \frac{2\pi}{2\pi} = 1 \).
4. Graph one period of the function. Start at \( x = 0 \) and end at \( x = 1 \), since the period is 1. Plot key points such as the maximum, minimum, and intercepts based on the amplitude and period.
5. Consider the negative sign in front of the amplitude, which indicates a reflection over the x-axis. This means the graph will start at the minimum point instead of the maximum.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of sine functions, it is determined by the coefficient in front of the sine term. For the function y = -3 sin 2πx, the amplitude is 3, indicating that the wave oscillates 3 units above and below the central axis.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For sine functions, the period can be calculated using the formula 2π divided by the coefficient of x inside the sine function. In this case, the period of y = -3 sin 2πx is 1, meaning the function completes one full cycle over the interval from x = 0 to x = 1.

Graphing Sine Functions

Graphing sine functions involves plotting the values of the function over a specified interval. For y = -3 sin 2πx, the graph will oscillate between -3 and 3, with the wave starting at the central axis, reaching its maximum at x = 0.25, returning to the axis at x = 0.5, reaching its minimum at x = 0.75, and completing the cycle at x = 1.