Mathematics
Grade 12
15 min
Write equations of sine functions using properties
Write equations of sine functions using properties
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1
Introduction & Learning Objectives
Learning Objectives
Identify the amplitude, period, phase shift, and vertical shift from the graph of a sinusoidal function.
Calculate the vertical shift (D) and amplitude (A) using the maximum and minimum values of the function.
Determine the period from a graph and use it to calculate the parameter B.
Identify the horizontal (phase) shift (C) by locating a starting point of the sine cycle.
Determine the sign of the amplitude (A) based on whether the function is increasing or decreasing at its starting point.
Synthesize the parameters A, B, C, and D to write a complete and accurate equation in the form y = A sin(B(x - C)) + D.
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Key Concepts & Vocabulary
TermDefinitionExample
AmplitudeThe amplitude is the distance from the function's central axis (midline) to its maximum or minimum value. It represents half the total vertical distance between the peak and trough and is always a positive value, denoted as |A|.If a sine wave has a maximum value of 5 and a minimum value of -1, its amplitude is (5 - (-1)) / 2 = 3.
PeriodThe period is the length of one complete cycle of the wave, measured along the horizontal axis. It is the horizontal distance after which the function's values begin to repeat.If a sine wave starts a cycle at x=0 and completes that cycle at x=4π, its period is 4π.
Phase ShiftThe phase shift is the horizontal displacement of the sine function from its parent function, y = sin(x). It indicates where a cycle begins.In t...
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Core Formulas
Standard Form of a Sine Function
y = A \sin(B(x - C)) + D
This is the standard transformed sine function equation. A controls the amplitude and vertical reflection. B controls the period. C controls the horizontal (phase) shift. D controls the vertical shift (the midline).
Calculating Parameters from Graph Properties
D = \frac{\text{Max} + \text{Min}}{2} \quad |A| = \frac{\text{Max} - \text{Min}}{2} \quad B = \frac{2\pi}{\text{Period}}
These three formulas are used to calculate the vertical shift (D), the amplitude (|A|), and the period-related parameter (B) directly from the maximum value, minimum value, and period observed on the graph.
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Challenging
A sinusoidal function passes through a minimum at (0, 1) and the next consecutive maximum at (π, 9). What is the equation of this function?
A.y = -4 sin(x - π/2) + 5
B.y = 4 sin(x - π/2) + 5
C.y = -4 cos(x) + 5
D.y = 4 cos(x - π) + 5
Challenging
Consider the function f(x) = -3 sin(2x + π/2) - 1. Which of the following statements is FALSE?
A.The range of the function is [-4, 2].
B.The period of the function is π.
C.The phase shift is π/2 units to the left.
D.The function has a y-intercept at (0, -4).
Challenging
A sine wave has the equation y = A sin(π/4(x - C)) + D. It has a maximum point at (5, 12) and the closest midline point with a positive slope is at (3, 7). What is the equation?
A.y = 5 sin(π/4(x - 3)) + 7
B.y = 5 sin(π/4(x - 5)) + 7
C.y = 12 sin(π/4(x - 3)) + 5
D.y = 7 sin(π/4(x - 3)) + 5
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