y=3sin (2x+ π/2)-1 graph and find the period, range,  exact domain

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.

y=3sin(2x+ π/2)-1 graph and find the period, range, exact domain

Explore how to graph the trigonometric function y = 3sin(2x + π/2) - 1 and determine its period, range, and exact domain. This detailed guide covers key trigonometric concepts and formulas, making it ideal for high school students in Grades 10-12.

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