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To graph the function [tex]\( y = 2 \sin \left( \frac{\pi}{10} x \right) - 7 \)[/tex], let's follow these steps:
### Step 1: Understand the Function
The function [tex]\( y = 2 \sin \left( \frac{\pi}{10} x \right) - 7 \)[/tex] consists of several components:
- The sine function [tex]\( \sin \left( \frac{\pi}{10} x \right) \)[/tex] is a periodic function.
- The coefficient 2 in front of the sine function scales the amplitude.
- The term [tex]\( -7 \)[/tex] shifts the entire function downward by 7 units.
### Step 2: Find the Amplitude, Period, and Vertical Shift
- Amplitude: The amplitude of [tex]\( \sin(x) \)[/tex] is 1, but since the sine function is multiplied by 2, the amplitude becomes 2. This means the graph will oscillate between 2 and -2 before any vertical shift.
- Vertical Shift: The term [tex]\( -7 \)[/tex] shifts the graph down by 7 units. So, instead of oscillating between 2 and -2, the graph will oscillate between [tex]\( -5 \)[/tex] and [tex]\( -9 \)[/tex].
- Period: The period of the sine function is determined by the coefficient inside the sine function. For [tex]\( \sin(bx) \)[/tex], the period is [tex]\( \frac{2 \pi}{b} \)[/tex]. Here [tex]\( b = \frac{\pi}{10} \)[/tex], so the period [tex]\( = \frac{2 \pi}{\frac{\pi}{10}} = 20 \)[/tex].
### Step 3: Identify Key Points
The sine function has key points at:
- [tex]\( \sin(0) = 0 \)[/tex]
- [tex]\( \sin\left(\frac{\pi}{2}\right) = 1 \)[/tex]
- [tex]\( \sin(\pi) = 0 \)[/tex]
- [tex]\( \sin\left(\frac{3\pi}{2}\right) = -1 \)[/tex]
- [tex]\( \sin(2\pi) = 0 \)[/tex]
Given the period of 20, these key points translate to:
- [tex]\( \sin\left( \frac{\pi}{10} x \right) = 0 \)[/tex] at [tex]\( x = 0, 20, 40, \ldots \)[/tex]
- [tex]\( \sin\left( \frac{\pi}{10} x \right) = 1 \)[/tex] at [tex]\( x = 10, 30, 50, \ldots \)[/tex]
- [tex]\( \sin\left( \frac{\pi}{10} x \right) = -1 \)[/tex] at [tex]\( x = 30, 70, \ldots \)[/tex]
Applying the amplitude and vertical shift, we find these points will contribute to:
- [tex]\( y = -7 \)[/tex] (midline) at [tex]\( x = 0, 20, 40, \ldots \)[/tex]
- [tex]\( y = -5 \)[/tex] (maximum) at [tex]\( x = 10, 30, 50, \ldots \)[/tex]
- [tex]\( y = -9 \)[/tex] (minimum) at [tex]\( x = 30, 70, \ldots \)[/tex]
### Step 4: Plot the Function
To graph [tex]\( y = 2 \sin \left( \frac{\pi}{10} x \right) - 7 \)[/tex]:
1. Vertical Shift: Begin by shifting the midline from [tex]\( y = 0 \)[/tex] to [tex]\( y = -7 \)[/tex].
2. Amplitude: Plot the amplitude oscillations from -5 to -9.
3. Period: Ensure the function completes one full cycle every 20 units along the x-axis.
### Step 5: Interactive Points
In the interactive widget:
- Move one point to an extremum, say at [tex]\( x = 10 \)[/tex] where [tex]\( y = -5 \)[/tex].
- Move the other point to an adjacent intersection with the midline, for example at [tex]\( x = 0 \)[/tex] or [tex]\( x = 20 \)[/tex] where [tex]\( y = -7 \)[/tex].
This will give a visual representation of the function along with the key feature points.
### Step 1: Understand the Function
The function [tex]\( y = 2 \sin \left( \frac{\pi}{10} x \right) - 7 \)[/tex] consists of several components:
- The sine function [tex]\( \sin \left( \frac{\pi}{10} x \right) \)[/tex] is a periodic function.
- The coefficient 2 in front of the sine function scales the amplitude.
- The term [tex]\( -7 \)[/tex] shifts the entire function downward by 7 units.
### Step 2: Find the Amplitude, Period, and Vertical Shift
- Amplitude: The amplitude of [tex]\( \sin(x) \)[/tex] is 1, but since the sine function is multiplied by 2, the amplitude becomes 2. This means the graph will oscillate between 2 and -2 before any vertical shift.
- Vertical Shift: The term [tex]\( -7 \)[/tex] shifts the graph down by 7 units. So, instead of oscillating between 2 and -2, the graph will oscillate between [tex]\( -5 \)[/tex] and [tex]\( -9 \)[/tex].
- Period: The period of the sine function is determined by the coefficient inside the sine function. For [tex]\( \sin(bx) \)[/tex], the period is [tex]\( \frac{2 \pi}{b} \)[/tex]. Here [tex]\( b = \frac{\pi}{10} \)[/tex], so the period [tex]\( = \frac{2 \pi}{\frac{\pi}{10}} = 20 \)[/tex].
### Step 3: Identify Key Points
The sine function has key points at:
- [tex]\( \sin(0) = 0 \)[/tex]
- [tex]\( \sin\left(\frac{\pi}{2}\right) = 1 \)[/tex]
- [tex]\( \sin(\pi) = 0 \)[/tex]
- [tex]\( \sin\left(\frac{3\pi}{2}\right) = -1 \)[/tex]
- [tex]\( \sin(2\pi) = 0 \)[/tex]
Given the period of 20, these key points translate to:
- [tex]\( \sin\left( \frac{\pi}{10} x \right) = 0 \)[/tex] at [tex]\( x = 0, 20, 40, \ldots \)[/tex]
- [tex]\( \sin\left( \frac{\pi}{10} x \right) = 1 \)[/tex] at [tex]\( x = 10, 30, 50, \ldots \)[/tex]
- [tex]\( \sin\left( \frac{\pi}{10} x \right) = -1 \)[/tex] at [tex]\( x = 30, 70, \ldots \)[/tex]
Applying the amplitude and vertical shift, we find these points will contribute to:
- [tex]\( y = -7 \)[/tex] (midline) at [tex]\( x = 0, 20, 40, \ldots \)[/tex]
- [tex]\( y = -5 \)[/tex] (maximum) at [tex]\( x = 10, 30, 50, \ldots \)[/tex]
- [tex]\( y = -9 \)[/tex] (minimum) at [tex]\( x = 30, 70, \ldots \)[/tex]
### Step 4: Plot the Function
To graph [tex]\( y = 2 \sin \left( \frac{\pi}{10} x \right) - 7 \)[/tex]:
1. Vertical Shift: Begin by shifting the midline from [tex]\( y = 0 \)[/tex] to [tex]\( y = -7 \)[/tex].
2. Amplitude: Plot the amplitude oscillations from -5 to -9.
3. Period: Ensure the function completes one full cycle every 20 units along the x-axis.
### Step 5: Interactive Points
In the interactive widget:
- Move one point to an extremum, say at [tex]\( x = 10 \)[/tex] where [tex]\( y = -5 \)[/tex].
- Move the other point to an adjacent intersection with the midline, for example at [tex]\( x = 0 \)[/tex] or [tex]\( x = 20 \)[/tex] where [tex]\( y = -7 \)[/tex].
This will give a visual representation of the function along with the key feature points.
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