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Sagot :
Sure, let's go through a detailed, step-by-step solution to graph the function [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) - 2 \)[/tex].
### Step 1: Understand the Base Cosine Function
The basic cosine function is [tex]\( y = \cos(x) \)[/tex]. It oscillates between 1 and -1, has a period of [tex]\(2\pi\)[/tex], and is symmetric about the y-axis.
### Step 2: Adjust for the Amplitude
The function we have is [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) - 2 \)[/tex].
- The amplitude of the base cosine function is multiplied by 6. This means the graph now oscillates between -6 and 6.
- So in our specific function, the peaks of the cosine curve will reach [tex]\(6 \cdot 1 = 6\)[/tex] and the troughs will reach [tex]\(6 \cdot (-1) = -6\)[/tex].
### Step 3: Adjust for the Vertical Shift
- The function [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) \)[/tex] will normally oscillate between 6 and -6. However, the function we have is [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) - 2 \)[/tex].
- [tex]\( -2 \)[/tex] is a vertical shift downwards. This means that all values of y will be decreased by 2:
- Maximum value: [tex]\(6 - 2 = 4\)[/tex]
- Minimum value: [tex]\(-6 - 2 = -8\)[/tex].
### Step 4: Adjust the Period
- The period of the basic cosine function [tex]\( y = \cos(x) \)[/tex] is [tex]\(2\pi\)[/tex].
- In our function, we have a modification inside the cosine term: [tex]\(y = 6 \cos \left(\frac{\pi}{4} x \right) - 2\)[/tex].
- The period of the function [tex]\( y = \cos(kx) \)[/tex] is [tex]\(\frac{2\pi}{k}\)[/tex]. Here, [tex]\(k = \frac{\pi}{4}\)[/tex], so:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 8. \][/tex]
### Graphing the Function
1. Middle Line: The middle of the oscillation is provided by the vertical shift, which is [tex]\(y = -2\)[/tex].
2. Amplitude: From the middle line [tex]\( y = -2 \)[/tex], the graph oscillates with an amplitude of 6 units up and down from this line:
- Upper Extremum: [tex]\( y = -2 + 6 = 4 \)[/tex]
- Lower Extremum: [tex]\( y = -2 - 6 = -8 \)[/tex].
3. Period:
- The period is 8, so one full cycle of the wave will take 8 units along the x-axis.
### Plot Important Points
1. Starting point at [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \cos (0) - 2 = 4 \)[/tex].
2. One-fourth period [tex]\( x = 2 \)[/tex], reach the midline at [tex]\( y = -2 \)[/tex].
3. Half period [tex]\( x = 4 \)[/tex], reach the minimum [tex]\( y = -8 \)[/tex].
4. Three-fourth period [tex]\( x = 6 \)[/tex], reach the midline at [tex]\( y = -2 \)[/tex].
5. Full period [tex]\( x = 8 \)[/tex], back to the maximum [tex]\( y = 4 \)[/tex].
With these key points:
- Peak at [tex]\( (0, 4) \)[/tex].
- Midline crossing at [tex]\( (2, -2) \)[/tex].
- Trough at [tex]\( (4, -8) \)[/tex].
- Midline crossing at [tex]\( (6, -2) \)[/tex].
- Returning to peak at [tex]\( (8, 4) \)[/tex].
This should help you accurately plot the graph [tex]\(y = 6 \cos \left(\frac{\pi}{4} x\right) - 2\)[/tex].
### Step 1: Understand the Base Cosine Function
The basic cosine function is [tex]\( y = \cos(x) \)[/tex]. It oscillates between 1 and -1, has a period of [tex]\(2\pi\)[/tex], and is symmetric about the y-axis.
### Step 2: Adjust for the Amplitude
The function we have is [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) - 2 \)[/tex].
- The amplitude of the base cosine function is multiplied by 6. This means the graph now oscillates between -6 and 6.
- So in our specific function, the peaks of the cosine curve will reach [tex]\(6 \cdot 1 = 6\)[/tex] and the troughs will reach [tex]\(6 \cdot (-1) = -6\)[/tex].
### Step 3: Adjust for the Vertical Shift
- The function [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) \)[/tex] will normally oscillate between 6 and -6. However, the function we have is [tex]\( y = 6 \cos \left(\frac{\pi}{4} x \right) - 2 \)[/tex].
- [tex]\( -2 \)[/tex] is a vertical shift downwards. This means that all values of y will be decreased by 2:
- Maximum value: [tex]\(6 - 2 = 4\)[/tex]
- Minimum value: [tex]\(-6 - 2 = -8\)[/tex].
### Step 4: Adjust the Period
- The period of the basic cosine function [tex]\( y = \cos(x) \)[/tex] is [tex]\(2\pi\)[/tex].
- In our function, we have a modification inside the cosine term: [tex]\(y = 6 \cos \left(\frac{\pi}{4} x \right) - 2\)[/tex].
- The period of the function [tex]\( y = \cos(kx) \)[/tex] is [tex]\(\frac{2\pi}{k}\)[/tex]. Here, [tex]\(k = \frac{\pi}{4}\)[/tex], so:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 8. \][/tex]
### Graphing the Function
1. Middle Line: The middle of the oscillation is provided by the vertical shift, which is [tex]\(y = -2\)[/tex].
2. Amplitude: From the middle line [tex]\( y = -2 \)[/tex], the graph oscillates with an amplitude of 6 units up and down from this line:
- Upper Extremum: [tex]\( y = -2 + 6 = 4 \)[/tex]
- Lower Extremum: [tex]\( y = -2 - 6 = -8 \)[/tex].
3. Period:
- The period is 8, so one full cycle of the wave will take 8 units along the x-axis.
### Plot Important Points
1. Starting point at [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \cos (0) - 2 = 4 \)[/tex].
2. One-fourth period [tex]\( x = 2 \)[/tex], reach the midline at [tex]\( y = -2 \)[/tex].
3. Half period [tex]\( x = 4 \)[/tex], reach the minimum [tex]\( y = -8 \)[/tex].
4. Three-fourth period [tex]\( x = 6 \)[/tex], reach the midline at [tex]\( y = -2 \)[/tex].
5. Full period [tex]\( x = 8 \)[/tex], back to the maximum [tex]\( y = 4 \)[/tex].
With these key points:
- Peak at [tex]\( (0, 4) \)[/tex].
- Midline crossing at [tex]\( (2, -2) \)[/tex].
- Trough at [tex]\( (4, -8) \)[/tex].
- Midline crossing at [tex]\( (6, -2) \)[/tex].
- Returning to peak at [tex]\( (8, 4) \)[/tex].
This should help you accurately plot the graph [tex]\(y = 6 \cos \left(\frac{\pi}{4} x\right) - 2\)[/tex].
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