In Exercises 75–78, graph one period of each function. y = −|3 sin πx|

Verified step by step guidance

1. Identify the base trigonometric function inside the absolute value, which is \(\sin(\pi x)\), and note that its period is given by \(\frac{2\pi}{B}\) where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).
2. Since the function is \(y = -|3 \sin(\pi x)|\), first consider the inner function \(3 \sin(\pi x)\). The amplitude of \(\sin(\pi x)\) is 1, so multiplying by 3 scales the amplitude to 3.
3. The absolute value \(|3 \sin(\pi x)|\) means all negative values of \(3 \sin(\pi x)\) become positive, reflecting any part of the graph below the x-axis to above the x-axis.
4. The negative sign outside the absolute value, \(- |3 \sin(\pi x)|\), flips the entire graph of \(|3 \sin(\pi x)|\) over the x-axis, making all values non-positive (zero or negative).
5. To graph one period, plot the function from \(x=0\) to \(x=2\), marking key points such as where \(\sin(\pi x)\) is zero, positive, or negative, apply the absolute value, then apply the negative sign, and connect these points smoothly to reflect the shape of the transformed sine wave.

Key Concepts

Period of a Sine Function

The period of a sine function is the length of one complete cycle on the x-axis. For y = sin(bx), the period is given by 2π / b. Understanding the period helps in determining the interval over which to graph one full cycle of the function.

Absolute Value Transformation

Applying the absolute value to a function, such as |f(x)|, reflects all negative values of f(x) above the x-axis, making the output non-negative. This changes the shape of the graph by 'folding' any parts below the x-axis upwards.

Vertical Scaling and Reflection

Multiplying a function by a constant scales it vertically, stretching or compressing the graph. A negative multiplier, like -|3 sin πx|, reflects the graph across the x-axis, flipping it upside down, which affects the amplitude and orientation of the graph.