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
To sketch the sine functions and determine their domain and range, follow these steps:
1. Understand the general form of the sine function: y = A sin (Bx + C) + D
- A represents the amplitude (the maximum vertical distance the wave reaches from the x-axis)
- B determines the period (the distance over which the function repeats)
- C represents the phase shift (the horizontal displacement of the graph)
- D is the vertical shift (the height added or subtracted from the x-axis)
2. For each given sine function, identify the values of A, B, C, and D.
3. Determine the domain of the function, which is typically all real numbers since the sine function extends indefinitely in both directions.
4. Calculate the period using the formula: Period = 2π / |B|
5. Sketch the graph by starting at the origin (0, 0) and marking the y-values at intervals of the period. Note the maximum and minimum points based on the amplitude.
6. Repeat the process to complete at least one full period of the sine function on the graph.
7. State the domain, which is all real numbers, and the range, which is the set of y-values that the function can take.
Now, using the above steps, let's sketch the sine functions and determine their domain and range:
Part A:
1) y = 3 sin (x)
- A = 3, B = 1 (since Bx implies a coefficient of 1), C = 0, D = 0
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [-3, 3] since the amplitude is 3.
2) y = 2 sin (x - 60°)
- A = 2, B = 1, C = -60°, D = 0
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [-2, 2] since the amplitude is 2.
3) y = - sin (x) + 2
- A = -1, B = 1, C = 0, D = 2
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [1, 3] since the amplitude is 1 and the vertical shift is 2.
4) y = -2 sin (x + 90°) + 3
- A = -2, B = 1, C = 90°, D = 3
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [1, 5] since the amplitude is 2 and the vertical shift is 3.
5) y = -3 sin (x - 20°) - 2
- A = -3, B = 1, C = -20°, D = -2
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [-5, -1] since the amplitude is 3 and the vertical shift is -2.
6) y = 1/3 sin (x)
- A = 1/3, B = 1, C = 0, D = 0
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [-1/3, 1/3] since the amplitude is 1/3.
7) y = 1/2 sin (x - 10°)
- A = 1/2, B = 1, C = -10°, D = 0
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [-1/2, 1/2] since the amplitude is 1/2.
8) y = 1/4 sin (x + 45°) - 1
- A = 1/4, B = 1, C = 45°, D = -1
- The domain is all real numbers.
- The period is 2π since |B| = 1.
- The range of y is [-1.25, -0.75] since the amplitude is 1/4 and the vertical shift is -1.
After following these steps for each sine function, you can sketch the graphs accordingly and state the domain and range for each function.
As an AI text-based bot, I cannot physically draw or submit sketches. However, I can provide you with a step-by-step guide on how to sketch these sine functions.
To sketch each function, you will need to plot several points and then connect them to create a smooth curve. Here is how you can do it:
1. Determine the domain and range:
- The domain of sine functions is all real numbers: (-∞, ∞).
- The range of sine functions is generally between -1 and 1 (inclusive), depending on any vertical shifts in the function.
2. Find the key points:
- The amplitude (A) is the distance from the midline to the maximum or minimum point of the function. It can be calculated as the absolute value of the coefficient of the sine function. For example, in the function y = 3 sin (x), the amplitude is 3.
- The period (P) of the sine function is the distance between two consecutive peak points (or trough points). It can be calculated using the formula P = 2π / b, where b is the coefficient of x in the sine function. For example, in the function y = 2 sin (x - 60°), the period is 2π.
3. Calculate additional points:
- Choose some x-values within one period (P) of the function and calculate the corresponding y-values using the given equation.
- You can use the coordinates (0, 0) as one point on the graph since sin(0) = 0.
4. Plot the points:
- Mark the x-values on the x-axis and the corresponding y-values on the y-axis.
- Repeat this process for several points within one period to get a good representation of the graph.
5. Sketch the curve:
- Connect the plotted points with a smooth curve, following the general shape of a sine function.
Remember to label the axes and any important points on the graph, such as maximum or minimum points and the midline.
Once you have completed the sketching, you can use any scanning or image-capturing tool to create an image file and submit it to your teacher.