Let's break down the function $y = 3 \sin (2x + \frac{\pi}{2}) - 1$, graph it, and analyze its properties, including the period, range, and domain.

Step 1: Rewrite the Function

The given function can be expressed in a standard sine form: $y = A \sin(Bx + C) + D$ Where:

• $A = 3$ (amplitude),
• $B = 2$ (affects the period),
• $C = \frac{\pi}{2}$ (horizontal shift or phase shift),
• $D = -1$ (vertical shift).

Step 2: Find the Period

The period of a sine function is calculated as: $\text{Period} = \frac{2\pi}{|B|}$ In this case, $B = 2$, so the period is: $\text{Period} = \frac{2\pi}{2} = \pi$ Thus, the period is $\pi$, meaning the function completes one full cycle over an interval of length $\pi$.

Step 3: Find the Range

The range of a sine function is determined by the amplitude and vertical shift. For $y = A \sin(Bx + C) + D$, the range is: $[\ D - A,\ D + A \ ]$ Substituting the values $A = 3$ and $D = -1$: [ \text{Range} = [-1 - 3, -1 + 3] = [-4, 2] ] Thus, the range of the function is $[-4, 2]$.

Step 4: Find the Exact Domain

Since the sine function is periodic and defined for all real numbers, the domain of the function is: $\text{Domain} = (-\infty, \infty)$

Step 5: Graph Description

• The amplitude is 3, so the graph oscillates between -4 and 2.
• The vertical shift moves the graph down by 1 unit.
• The phase shift is determined by $C = \frac{\pi}{2}$. To find the phase shift, we calculate: $\text{Phase Shift} = -\frac{C}{B} = -\frac{\frac{\pi}{2}}{2} = -\frac{\pi}{4}$ So, the graph is shifted to the left by $\frac{\pi}{4}$.
• The period is $\pi$, so the graph repeats every $\pi$ units.

Summary

• Period: $\pi$
• Range: $[-4, 2]$
• Domain: $(-\infty, \infty)$

Would you like me to plot the graph or provide more details? Here are some further questions that build on this:

1. How does the vertical shift $D$ affect the overall appearance of the graph?
2. What happens to the graph if the phase shift $C$ changes?
3. How does changing the amplitude $A$ modify the sine wave's behavior?
4. What is the relationship between the period and the frequency of a sine wave?
5. How do we determine key points for sketching a sine graph manually?

Tip: For transformations of trigonometric functions, always address amplitude, period, phase shift, and vertical shift in that order for clarity.