Mathematics
Grade 11
15 min
Write equations of sine functions from graphs
Write equations of sine functions from graphs
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1
Introduction & Learning Objectives
Learning Objectives
Identify the amplitude, period, midline, and phase shift of a sine function from its graph.
Calculate the vertical shift (d) and amplitude (a) using the maximum and minimum values of the graph.
Determine the period from the graph and use it to calculate the frequency parameter (b).
Identify an appropriate horizontal phase shift (c) for a sine function.
Synthesize the parameters a, b, c, and d into a complete equation of the form y = a sin(b(x - c)) + d.
Determine the correct sign of the amplitude (a) based on whether the function is increasing or decreasing at its starting point.
Verify their final equation by substituting a known point from the graph.
Have you ever wondered how sound engineers model sound waves or how astronomers predict the hours of d...
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Key Concepts & Vocabulary
TermDefinitionExample
MidlineThe horizontal line that passes exactly halfway between the graph's maximum and minimum points. It represents the vertical shift.If a graph's maximum is y=5 and its minimum is y=1, the midline is y = (5+1)/2 = 3.
Amplitude (a)The distance from the midline to either the maximum or minimum point of the graph. It represents the vertical stretch or compression and is always a positive value.For a graph with a maximum of y=5 and a midline of y=3, the amplitude is 5 - 3 = 2.
PeriodThe length of one full cycle of the wave, measured along the x-axis. It is the horizontal distance before the graph starts to repeat itself.If a sine wave starts a cycle at x=0 and completes it at x=4π, the period is 4π.
Vertical Shift (d)The value that indicates how far the grap...
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Core Formulas
General Equation of a Sine Function
y = a \sin(b(x - c)) + d
This is the standard form used to represent a transformed sine function. 'a' controls amplitude and reflection, 'b' controls the period, 'c' controls the phase shift, and 'd' controls the vertical shift.
Formulas for Parameters from a Graph
d = \frac{Max + Min}{2} \quad |a| = \frac{Max - Min}{2} \quad |b| = \frac{2\pi}{Period}
Use these formulas to calculate the vertical shift (d), amplitude (a), and the frequency parameter (b) directly from the maximum, minimum, and period values identified on the graph.
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Challenging
The graph of y = 2sin(x) is shown. All of the following equations represent the same graph EXCEPT:
A.y = 2sin(x - 2π)
B.y = -2sin(x - π)
C.y = 2sin(x + 2π)
D.y = 2sin(x - π)
Challenging
A sine function has an amplitude of 3, a period of π, and a midline of y = -1. The function's graph passes through the point (π/4, 2). Which of the following is a possible equation for this function?
A.y = 3sin(2x) - 1
B.y = -3sin(2x) - 1
C.y = 3sin(x - π/4) - 1
D.y = 3sin(2(x - π/2)) - 1
Challenging
Consider the functions f(x) = 4sin(x - π/2) and g(x) = 4sin(x + π/2). How is the graph of g(x) related to the graph of f(x)?
A.g(x) is f(x) shifted right by π
B.g(x) is f(x) shifted left by π/2
C.g(x) is f(x) shifted left by π
D.g(x) is a reflection of f(x) over the y-axis
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