Question

Graph the given function over one period. $$y=-3 \sin (\pi x)$$

   Graph the given function over one period.
$$y=-3 \sin (\pi x)$$

Precalculus

Cynthia Young 2nd Edition

Chapter 5, Problem 27

Instant Answer

Solved by Expert Robert Leedy

06/14/2020

Step 1

The amplitude is the absolute value of the coefficient of the sine function, which is 3. The period is the reciprocal of the absolute value of the coefficient of x, which is $\frac{2\pi}{\pi}=2$. The phase shift is the value that is subtracted from x inside the Show more…

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Graph the given function over one period. $$y=-3 \sin (\pi x)$$

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Key Concepts

Graph Transformations

Graph transformations encompass various modifications of basic functions, such as translations, reflections, stretches, and compressions. Mastery of these transformations is essential for accurately sketching and understanding the behavior of functions, especially in the context of trigonometric graphs.

Sine Function

The sine function is one of the primary trigonometric functions, characterized by its smooth, periodic oscillations. It serves as a fundamental building block in mathematics for modeling cyclic and wave-like phenomena.

Amplitude

Amplitude is the measure of the maximum displacement of a wave from its central axis. In trigonometric functions, it determines the scale of the vertical stretch or compression, indicating how far the peaks and troughs of the wave extend from the midline.

Period

The period is the length of one complete cycle of a periodic function. For sine functions, any coefficient multiplying the variable inside the sine function affects the period, resulting in a horizontal compression or stretching of the graph.

Vertical Reflection

Vertical reflection occurs when the function is multiplied by a negative factor, which flips the graph over the horizontal axis. This transformation alters the orientation of the waveform, making peaks become troughs and vice versa.

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