To graph the function [tex]\( y = -\sin \left( \frac{\pi}{8} x \right) + 7 \)[/tex], let's break it down step-by-step:
1. Identify the Midline: The midline of the function is determined by the vertical shift. For the function [tex]\( y = -\sin \left( \frac{\pi}{8} x \right) + 7 \)[/tex], the midline is [tex]\( y = 7 \)[/tex].
2. Find the Amplitude: The amplitude of the function is the coefficient in front of the sine function. Here, the amplitude is 1 (since the coefficient is 1, with the negative sign indicating a reflection across the x-axis).
3. Determine the Period: The period of a sine function is calculated using the formula [tex]\( \frac{2\pi}{b} \)[/tex], where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the sine function.
[tex]\[
\text{Period} = \frac{2\pi}{\frac{\pi}{8}} = 16
\][/tex]
4. Phase Shift and Vertical Shift:
- There is no phase shift in this function since there is no horizontal shift term inside the sine function.
- The vertical shift moves the entire graph upwards by 7 units.
5. Critical Points:
- Maximum Point: Since the sine function has been reflected, it reaches its maximum at the midline plus the amplitude: [tex]\( 7 + 1 = 8 \)[/tex].
- Minimum Point: The sine function's minimum is at the midline minus the amplitude: [tex]\( 7 - 1 = 6 \)[/tex].
6. Graphing One Cycle:
- Start at [tex]\( x = 0, y = 7 \)[/tex]: This is the midline.
- At [tex]\( x = 4, y = 8 \)[/tex]: This is a maximum point.
- At [tex]\( x = 8, y = 7 \)[/tex]: Back to the midline.
- At [tex]\( x = 12, y = 6 \)[/tex]: This is a minimum point.
- At [tex]\( x = 16, y = 7 \)[/tex]: Back to the midline completing one period of 16 units.
Now let's plot these points:
- Start with the midline at [tex]\( y = 7 \)[/tex].
- Identify the points [tex]\( (0, 7) \)[/tex], [tex]\( (4, 8) \)[/tex], [tex]\( (8, 7) \)[/tex], [tex]\( (12, 6) \)[/tex], and [tex]\( (16, 7) \)[/tex].
To reflect this function correctly, these points help form the sinusoidal shape with respect to the midline. You can connect these points smoothly to represent the sine wave.
Practice plotting these critical points and complete the sine curve to accurately graph the function [tex]\( y = -\sin \left( \frac{\pi}{8} x \right) + 7 \)[/tex].
