Create sketches by hand of the following sine functions, showing at least one full period and state the domain and range. Submit them to your teacher using the drop box above. It may be easiest for you to scan your hand drawn sketches and submit the scan file.
Part A: Sketching
y = 3 sin (x)
y = 2 sin (x - 60°)
y = - sin (x) + 2
y = -2 sin (x + 90°) + 3
y = -3 sin (x - 20°) - 2
y = 1 over 3 sin(x)
y = 1 over 2 sin(x - 10°)
y = 1 over 4 sin(x + 45°) - 1
enter your functions at wolframalpha.com and you will get graphs and other info.
use "/" and not "over" when typing your math. For example,
https://www.wolframalpha.com/input/?i=y+%3D+-2+sin+%28x+%2B+90%C2%B0%29+%2B+3
As an AI text-based bot, I cannot directly create sketches or submit files to your teacher. However, I can help you understand how to sketch these sine functions by hand and provide a verbal description of the sketches.
To sketch a sine function, follow these steps:
1. Determine the period: The period is the length of one complete cycle of the sine function. It can be found using the formula 2π/b, where b is the coefficient of x in the function.
2. Mark the x-axis: Label the x-axis from -2π to 2π or any appropriate multiple of the period, depending on the range of values you want to show in your sketch.
3. Mark the y-axis: Determine the amplitude, which is the coefficient of the sine function (excluding any fractional or negative values). Draw a horizontal line representing the amplitude.
4. Plot key points: Determine the vertical shifts, phase shifts, and any other transformations by comparing the function to the standard form y = A sin (Bx - C) + D.
- Vertical shift: If D is included, move the entire graph upward or downward by D units.
- Amplitude and vertical stretch/compression: If A is included, multiply the amplitude line by A.
- Phase shift: If C is included, shift the graph horizontally to the right (if C > 0) or left (if C < 0) by |C| units.
5. Sketch the graph: Connect the key points smoothly to create a smooth curve. Repeat the pattern for multiple periods if necessary.
Now, let's go through each of the sine functions you provided and describe their sketches verbally:
Part A:
1. y = 3 sin (x):
- Domain: The domain is all real numbers.
- Range: The range is [-3, 3], as the amplitude is 3.
- Sketch: It is a standard sine wave with an amplitude of 3.
2. y = 2 sin (x - 60°):
- Domain: The domain is all real numbers.
- Range: The range is [-2, 2], as the amplitude is 2.
- Sketch: It is a standard sine wave, but shifted horizontally 60° to the right.
3. y = - sin (x) + 2:
- Domain: The domain is all real numbers.
- Range: The range is [1, 3], as the amplitude is 1 and the graph is shifted vertically by 2 units.
- Sketch: It is an inverted sine wave, with an amplitude of 1 and shifted vertically 2 units upward.
4. y = -2 sin (x + 90°) + 3:
- Domain: The domain is all real numbers.
- Range: The range is [1, 5], as the amplitude is 2 and the graph is shifted vertically by 3 units.
- Sketch: It is an inverted sine wave, with an amplitude of 2, shifted horizontally 90° to the left, and shifted vertically 3 units upward.
5. y = -3 sin (x - 20°) - 2:
- Domain: The domain is all real numbers.
- Range: The range is [-5, -1], as the amplitude is 3 and the graph is shifted vertically by -2 units.
- Sketch: It is an inverted sine wave, with an amplitude of 3, shifted horizontally 20° to the right, and shifted vertically 2 units downward.
6. y = 1/3 sin(x):
- Domain: The domain is all real numbers.
- Range: The range is [-1/3, 1/3], as the amplitude is 1/3.
- Sketch: It is a standard sine wave with an amplitude of 1/3.
7. y = 1/2 sin(x - 10°):
- Domain: The domain is all real numbers.
- Range: The range is [-1/2, 1/2], as the amplitude is 1/2.
- Sketch: It is a standard sine wave, but shifted horizontally 10° to the right.
8. y = 1/4 sin(x + 45°) - 1:
- Domain: The domain is all real numbers.
- Range: The range is [-1.25, -0.75], as the amplitude is 1/4 and the graph is shifted vertically by -1 unit.
- Sketch: It is a standard sine wave, shifted horizontally 45° to the left, and shifted vertically 1 unit downward.
Remember, it's always best to double-check your sketches and make sure they match the given functions and the verbal descriptions. If possible, try to submit your scanned hand-drawn sketches using the appropriate drop box provided by your teacher.