Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Certainly! Let's construct and graph the sine function step by step, given the parameters:
1. Amplitude (A): The amplitude is the distance from the midline to the maximum or minimum value of the function. Here, the amplitude is 5.
2. Period (B): The period of the sine function is the distance (in the x-axis) over which the function completes one full cycle. It's given as \(6\pi\). The formula for period \(T\) of the function \(y = A \sin(Bx)\) is \(T = \frac{2\pi}{|B|}\):
[tex]\[ 6\pi = \frac{2\pi}{|B|} \][/tex]
Solving for \(B\):
[tex]\[ |B| = \frac{2\pi}{6\pi} = \frac{1}{3} \][/tex]
3. Midline (D): The midline is the horizontal axis around which the sine function oscillates. Here, it’s given as -2. So, \(D = -2\).
4. Reflection: The graph is a reflection over the x-axis, which means we multiply the sine function by -1. If the original sine function is \(y = \sin(x)\), the reflected function is \(y = -\sin(x)\).
Considering all of these together, our function takes the form:
[tex]\[ y = -5 \sin\left(\frac{1}{3}x\right) - 2 \][/tex]
To graph this function, follow these steps:
### Finding Key Points:
1. Y-Intercept: At \(x = 0\),
[tex]\[ y = -5 \sin\left(\frac{1}{3} \cdot 0\right) - 2 = -5 \sin(0) - 2 = -2 \][/tex]
So the point (0, -2) is on the midline.
2. Next Key Point (Minimum or Maximum):
The function is:
[tex]\[ y = -5 \sin\left(\frac{1}{3}x\right) - 2 \][/tex]
The minimum value of \(\sin(x)\) is -1 and the maximum value is 1. For this reflected and shifted function:
- Maximum: When \(\sin\left(\frac{1}{3}x\right) = -1\),
[tex]\[ y = -5(-1) - 2 = 5 - 2 = 3 \][/tex]
- Minimum: When \(\sin\left(\frac{1}{3}x\right) = 1\),
[tex]\[ y = -5(1) - 2 = -5 - 2 = -7 \][/tex]
3. Period and Quarter Points:
- Quarter points are intervals at \(\frac{period}{4}\). Period is \(6\pi\), therefore \(\frac{6\pi}{4} = \frac{3\pi}{2}\).
At each quarter interval from \(x = 0\):
- \(x = 0\): \(y = -2\)
- \(x = \frac{3\pi}{2}\): \(y\)-minimum \((=-7)\)
- \(x = 3\pi\): \(y = -2\)
- \(x = \frac{9\pi}{2}\): \(y\)-maximum \((=3)\)
- \(x = 6\pi\): \(y = -2\) (one full period)
### Plotting Points
1. Plot \((0, -2)\)
2. Plot \(\left(\frac{3\pi}{2}, -7\right)\)
3. Plot \((3\pi, -2)\)
4. Plot \(\left(\frac{9\pi}{2}, 3\right)\)
5. Plot \((6\pi, -2)\)
### Sketch the Curve
Connect these points smoothly to form the sinusoidal curve.
Using the sine tool, you can plot the function by placing your first point at (0, -2), and your close point for maximum or minimum nearby, following the instructions provided.
1. Amplitude (A): The amplitude is the distance from the midline to the maximum or minimum value of the function. Here, the amplitude is 5.
2. Period (B): The period of the sine function is the distance (in the x-axis) over which the function completes one full cycle. It's given as \(6\pi\). The formula for period \(T\) of the function \(y = A \sin(Bx)\) is \(T = \frac{2\pi}{|B|}\):
[tex]\[ 6\pi = \frac{2\pi}{|B|} \][/tex]
Solving for \(B\):
[tex]\[ |B| = \frac{2\pi}{6\pi} = \frac{1}{3} \][/tex]
3. Midline (D): The midline is the horizontal axis around which the sine function oscillates. Here, it’s given as -2. So, \(D = -2\).
4. Reflection: The graph is a reflection over the x-axis, which means we multiply the sine function by -1. If the original sine function is \(y = \sin(x)\), the reflected function is \(y = -\sin(x)\).
Considering all of these together, our function takes the form:
[tex]\[ y = -5 \sin\left(\frac{1}{3}x\right) - 2 \][/tex]
To graph this function, follow these steps:
### Finding Key Points:
1. Y-Intercept: At \(x = 0\),
[tex]\[ y = -5 \sin\left(\frac{1}{3} \cdot 0\right) - 2 = -5 \sin(0) - 2 = -2 \][/tex]
So the point (0, -2) is on the midline.
2. Next Key Point (Minimum or Maximum):
The function is:
[tex]\[ y = -5 \sin\left(\frac{1}{3}x\right) - 2 \][/tex]
The minimum value of \(\sin(x)\) is -1 and the maximum value is 1. For this reflected and shifted function:
- Maximum: When \(\sin\left(\frac{1}{3}x\right) = -1\),
[tex]\[ y = -5(-1) - 2 = 5 - 2 = 3 \][/tex]
- Minimum: When \(\sin\left(\frac{1}{3}x\right) = 1\),
[tex]\[ y = -5(1) - 2 = -5 - 2 = -7 \][/tex]
3. Period and Quarter Points:
- Quarter points are intervals at \(\frac{period}{4}\). Period is \(6\pi\), therefore \(\frac{6\pi}{4} = \frac{3\pi}{2}\).
At each quarter interval from \(x = 0\):
- \(x = 0\): \(y = -2\)
- \(x = \frac{3\pi}{2}\): \(y\)-minimum \((=-7)\)
- \(x = 3\pi\): \(y = -2\)
- \(x = \frac{9\pi}{2}\): \(y\)-maximum \((=3)\)
- \(x = 6\pi\): \(y = -2\) (one full period)
### Plotting Points
1. Plot \((0, -2)\)
2. Plot \(\left(\frac{3\pi}{2}, -7\right)\)
3. Plot \((3\pi, -2)\)
4. Plot \(\left(\frac{9\pi}{2}, 3\right)\)
5. Plot \((6\pi, -2)\)
### Sketch the Curve
Connect these points smoothly to form the sinusoidal curve.
Using the sine tool, you can plot the function by placing your first point at (0, -2), and your close point for maximum or minimum nearby, following the instructions provided.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
