Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = cos(x + π/2)

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 identifying the coefficient $a$. In this case, $a=1$, so the amplitude is $|1|=1$.
3. Find the period of the function using the formula $\frac{2\pi }{b}$. Here, $b=1$, so the period is $\frac{2\pi }{1}=2\pi$.
4. Calculate the phase shift using $-\frac{c}{b}$. With $c=\pi /2$ and $b=1$, the phase shift is $-\frac{\pi /2}{1}=-\pi /2$. This means the graph shifts $\pi /2$ units to the left.
5. Graph one period of the function by starting at the phase shift $-\pi /2$ and ending at $-\pi /2+2\pi$, marking key points such as maximum, minimum, and intercepts based on the amplitude and period.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of trigonometric functions like cosine, the amplitude is determined by the coefficient in front of the cosine term. For the function y = cos(x + π/2), the amplitude is 1, indicating that the graph oscillates between 1 and -1.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the cosine function, the standard period is 2π. In the function y = cos(x + π/2), there are no coefficients affecting the x variable, so the period remains 2π, meaning the function will repeat its values 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 from the x variable inside the function. In y = cos(x + π/2), the phase shift is -π/2, indicating that the graph is shifted π/2 units to the left along the x-axis.