Answer :
To determine the graph of the function [tex]\( h(x) = 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex], we need to analyze its key features and then describe how they manifest in the graph.
### Step-by-Step Analysis
1. Understanding the Function Transformation:
- The base function is [tex]\( \sin(x) \)[/tex].
- Adding [tex]\( \frac{\pi}{2} \)[/tex] to [tex]\( x \)[/tex] in [tex]\( \sin(x + \frac{\pi}{2}) \)[/tex] shifts the sine function to the left by [tex]\( \frac{\pi}{2} \)[/tex].
- Multiplying by 2 yields [tex]\( 2 \sin(x + \frac{\pi}{2}) \)[/tex], which scales the amplitude from 1 to 2.
- Subtracting 1 translates the function downward by 1 unit: [tex]\( 2 \sin(x + \frac{\pi}{2}) - 1 \)[/tex].
2. Key Characteristics of the Function:
- Amplitude: The amplitude is 2 (as given by the coefficient 2 in front of the sine function).
- Period: The period of [tex]\( \sin(x) \)[/tex] is [tex]\( 2\pi \)[/tex], so the period of [tex]\( 2 \sin(x + \frac{\pi}{2}) \)[/tex] is also [tex]\( 2\pi \)[/tex].
- Vertical Shift: The function is shifted downward by 1 unit.
- Phase Shift: The function is shifted to the left by [tex]\( \frac{\pi}{2} \)[/tex].
3. Important Points:
- The maximum value of [tex]\( \sin(x) \)[/tex] is 1, so the maximum value of [tex]\( 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex] is [tex]\( 2 \times 1 - 1 = 1 \)[/tex].
- The minimum value of [tex]\( \sin(x) \)[/tex] is -1, so the minimum value of [tex]\( 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex] is [tex]\( 2 \times (-1) - 1 = -3 \)[/tex].
4. Plotting the Function:
- Generate [tex]\( x \)[/tex] values over the interval from [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex], since [tex]\( 2\pi \)[/tex] is the period and we want to see at least one full cycle on each side.
- Compute the corresponding [tex]\( y \)[/tex] values using the function [tex]\( h(x) = 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex].
- Plot the points and connect them smoothly to form the sinusoidal wave.
### Graph Description
- Interval: From [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex].
- Key Points:
- At [tex]\( x = -2\pi \)[/tex], [tex]\( h(x) \approx 1 \)[/tex].
- At [tex]\( x = -\frac{3\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (one complete period forward).
- At [tex]\( x = -\pi \)[/tex], [tex]\( h(x) \approx -3 \)[/tex] (minimum value).
- At [tex]\( x = -\frac{\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (back to the midline after one half period).
- At [tex]\( x = 0 \)[/tex], [tex]\( h(x) = 1 \)[/tex] (maximum value).
- At [tex]\( x = \frac{\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (back to the midline after another quarter period).
- At [tex]\( x = \pi \)[/tex], [tex]\( h(x) \approx -3 \)[/tex] (minimum value).
- At [tex]\( x = \frac{3\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (back to the midline).
- At [tex]\( x = 2\pi \)[/tex], [tex]\( h(x) \approx 1 \)[/tex] (at the end of the second period).
These key points and their corresponding [tex]\( y \)[/tex]-values, along with the calculated range from -3 to 1, help visualize the characteristic sinusoidal wave:
The graph of [tex]\( h(x) = 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex] will complete one sinusoidal cycle within each [tex]\( 2\pi \)[/tex] interval, fluctuating between the peaks of 1 and -3 and crossing the midline at -1.
This detailed description matches the plotted points and thus describes the graph representing the function [tex]\( h(x) \)[/tex].
### Step-by-Step Analysis
1. Understanding the Function Transformation:
- The base function is [tex]\( \sin(x) \)[/tex].
- Adding [tex]\( \frac{\pi}{2} \)[/tex] to [tex]\( x \)[/tex] in [tex]\( \sin(x + \frac{\pi}{2}) \)[/tex] shifts the sine function to the left by [tex]\( \frac{\pi}{2} \)[/tex].
- Multiplying by 2 yields [tex]\( 2 \sin(x + \frac{\pi}{2}) \)[/tex], which scales the amplitude from 1 to 2.
- Subtracting 1 translates the function downward by 1 unit: [tex]\( 2 \sin(x + \frac{\pi}{2}) - 1 \)[/tex].
2. Key Characteristics of the Function:
- Amplitude: The amplitude is 2 (as given by the coefficient 2 in front of the sine function).
- Period: The period of [tex]\( \sin(x) \)[/tex] is [tex]\( 2\pi \)[/tex], so the period of [tex]\( 2 \sin(x + \frac{\pi}{2}) \)[/tex] is also [tex]\( 2\pi \)[/tex].
- Vertical Shift: The function is shifted downward by 1 unit.
- Phase Shift: The function is shifted to the left by [tex]\( \frac{\pi}{2} \)[/tex].
3. Important Points:
- The maximum value of [tex]\( \sin(x) \)[/tex] is 1, so the maximum value of [tex]\( 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex] is [tex]\( 2 \times 1 - 1 = 1 \)[/tex].
- The minimum value of [tex]\( \sin(x) \)[/tex] is -1, so the minimum value of [tex]\( 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex] is [tex]\( 2 \times (-1) - 1 = -3 \)[/tex].
4. Plotting the Function:
- Generate [tex]\( x \)[/tex] values over the interval from [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex], since [tex]\( 2\pi \)[/tex] is the period and we want to see at least one full cycle on each side.
- Compute the corresponding [tex]\( y \)[/tex] values using the function [tex]\( h(x) = 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex].
- Plot the points and connect them smoothly to form the sinusoidal wave.
### Graph Description
- Interval: From [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex].
- Key Points:
- At [tex]\( x = -2\pi \)[/tex], [tex]\( h(x) \approx 1 \)[/tex].
- At [tex]\( x = -\frac{3\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (one complete period forward).
- At [tex]\( x = -\pi \)[/tex], [tex]\( h(x) \approx -3 \)[/tex] (minimum value).
- At [tex]\( x = -\frac{\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (back to the midline after one half period).
- At [tex]\( x = 0 \)[/tex], [tex]\( h(x) = 1 \)[/tex] (maximum value).
- At [tex]\( x = \frac{\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (back to the midline after another quarter period).
- At [tex]\( x = \pi \)[/tex], [tex]\( h(x) \approx -3 \)[/tex] (minimum value).
- At [tex]\( x = \frac{3\pi}{2} \)[/tex], [tex]\( h(x) = -1 \)[/tex] (back to the midline).
- At [tex]\( x = 2\pi \)[/tex], [tex]\( h(x) \approx 1 \)[/tex] (at the end of the second period).
These key points and their corresponding [tex]\( y \)[/tex]-values, along with the calculated range from -3 to 1, help visualize the characteristic sinusoidal wave:
The graph of [tex]\( h(x) = 2 \sin \left(x + \frac{\pi}{2}\right) - 1 \)[/tex] will complete one sinusoidal cycle within each [tex]\( 2\pi \)[/tex] interval, fluctuating between the peaks of 1 and -3 and crossing the midline at -1.
This detailed description matches the plotted points and thus describes the graph representing the function [tex]\( h(x) \)[/tex].
