Matogo Pietha
2 days ago
Sine and cosine, to draw phase shift and vertical shift.
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ChatGPT
2 days ago
To draw the graphs of sine and cosine functions with phase shift and vertical shift, we need to understand how these shifts affect the standard graphs of these functions.
The standard graph of the sine function is a wave that starts at the origin, reaches a peak at (π/2, 1), crosses the x-axis at (π, 0), reaches a minimum at (3π/2, -1), and returns to the origin at (2π, 0). The standard graph of the cosine function is a wave that starts at (0, 1), reaches a minimum at (π, -1), crosses the x-axis at (π/2, 0), and returns to the starting point at (2π, 1).
To introduce a phase shift to the sine or cosine function, we simply shift the entire graph horizontally by a certain amount. For example, if we have a phase shift of π/4 for the sine function, we would shift the entire graph to the right by π/4 units. This means that the peak of the sine wave would now occur at (π/2 + π/4, 1) = (3π/4, 1), and the x-intercept would occur at (π + π/4, 0) = (5π/4, 0).
To introduce a vertical shift to the sine or cosine function, we simply shift the entire graph vertically by a certain amount. For example, if we have a vertical shift of 2 units for the cosine function, we would shift the entire graph up by 2 units. This means that the minimum of the cosine wave would now occur at (π, -1 + 2) = (π, 1), and the starting point would occur at (0, 1 + 2) = (0, 3).
By combining phase shifts and vertical shifts, we can create a variety of different graphs for sine and cosine functions.
The standard graph of the sine function is a wave that starts at the origin, reaches a peak at (π/2, 1), crosses the x-axis at (π, 0), reaches a minimum at (3π/2, -1), and returns to the origin at (2π, 0). The standard graph of the cosine function is a wave that starts at (0, 1), reaches a minimum at (π, -1), crosses the x-axis at (π/2, 0), and returns to the starting point at (2π, 1).
To introduce a phase shift to the sine or cosine function, we simply shift the entire graph horizontally by a certain amount. For example, if we have a phase shift of π/4 for the sine function, we would shift the entire graph to the right by π/4 units. This means that the peak of the sine wave would now occur at (π/2 + π/4, 1) = (3π/4, 1), and the x-intercept would occur at (π + π/4, 0) = (5π/4, 0).
To introduce a vertical shift to the sine or cosine function, we simply shift the entire graph vertically by a certain amount. For example, if we have a vertical shift of 2 units for the cosine function, we would shift the entire graph up by 2 units. This means that the minimum of the cosine wave would now occur at (π, -1 + 2) = (π, 1), and the starting point would occur at (0, 1 + 2) = (0, 3).
By combining phase shifts and vertical shifts, we can create a variety of different graphs for sine and cosine functions.
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Google Gemini AI
2 days ago
Sine Function
- General equation: y = A sin(Bx + C) + D
Phase Shift
- C: Shifts the graph horizontally by C/B units.
- If C is positive, the graph shifts left by C/B units.
- If C is negative, the graph shifts right by C/B units.
Vertical Shift
- D: Shifts the graph vertically by D units.
- If D is positive, the graph shifts up by D units.
- If D is negative, the graph shifts down by D units.
Cosine Function
- General equation: y = A cos(Bx + C) + D
Phase Shift
- C: Shifts the graph horizontally by C/B units.
- If C is positive, the graph shifts left by C/B units.
- If C is negative, the graph shifts right by C/B units.
Vertical Shift
- D: Shifts the graph vertically by D units.
- If D is positive, the graph shifts up by D units.
- If D is negative, the graph shifts down by D units.
- General equation: y = A sin(Bx + C) + D
Phase Shift
- C: Shifts the graph horizontally by C/B units.
- If C is positive, the graph shifts left by C/B units.
- If C is negative, the graph shifts right by C/B units.
Vertical Shift
- D: Shifts the graph vertically by D units.
- If D is positive, the graph shifts up by D units.
- If D is negative, the graph shifts down by D units.
Cosine Function
- General equation: y = A cos(Bx + C) + D
Phase Shift
- C: Shifts the graph horizontally by C/B units.
- If C is positive, the graph shifts left by C/B units.
- If C is negative, the graph shifts right by C/B units.
Vertical Shift
- D: Shifts the graph vertically by D units.
- If D is positive, the graph shifts up by D units.
- If D is negative, the graph shifts down by D units.
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