In Exercises 7–11, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = −3 cos (x + π)

Verified step by step guidance

1. Identify the standard form of the cosine function: \( y = a \cos(bx + c) + d \).
2. Determine the amplitude by taking the absolute value of \( a \). Here, \( a = -3 \), so the amplitude is \( |a| = 3 \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). In this case, \( b = 1 \), so the period is \( 2\pi \).
4. Calculate the phase shift using \( \frac{-c}{b} \). Here, \( c = \pi \) and \( b = 1 \), so the phase shift is \( -\pi \).
5. Graph one period of the function by starting at the phase shift \( -\pi \) and using the amplitude and period to plot the cosine wave.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of cosine functions, it is determined by the coefficient in front of the cosine term. For the function y = -3 cos(x + π), the amplitude is 3, indicating how far the graph stretches vertically from its midline.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For cosine functions, the standard period is 2π, but it can be altered by a coefficient in front of the variable x. In this case, since there is no coefficient affecting x, the period remains 2π, meaning the function will repeat every 2π units along the x-axis.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric function from its standard position. It is determined by the value added or subtracted inside the function's argument. For the function y = -3 cos(x + π), the phase shift is -π, indicating that the graph is shifted π units to the left along the x-axis.