Graph sine functions

Learning Objectives Identify the amplitude, period, phase shift, and vertical shift from the equation of a sine function. Graph the parent function y = sin(x) and identify its key features. Graph transformations of the sine function in the form y = A sin(B(x - C)) + D. Determine the equation of a sinusoidal function given its graph. Analyze the effect of each parameter (A, B, C, D) on the graph of y = sin(x). Determine the domain and range of any transformed sine function. Apply the five-point plotting method to accurately sketch one cycle of a sine function. Ever wondered how sound waves are visualized or how we model daily temperature cycles? 🌊 It all starts with the elegant, repeating curve of the sine function. This tutorial will guide you through the process of grap...

TermDefinitionExample AmplitudeHalf the vertical distance between the maximum and minimum values of the function. It represents the 'height' of the wave from its central axis and is given by |A|.For the function y = 4 sin(x), the amplitude is 4. The graph oscillates between y = -4 and y = 4. PeriodThe length of one complete horizontal cycle of the graph. It is the distance over which the function's shape repeats.For y = sin(2x), the period is 2π/2 = π. The graph completes one full wave over an interval of π radians. Phase ShiftThe horizontal translation of the sinusoidal graph. In the form y = A sin(B(x - C)) + D, the phase shift is C. A positive C shifts the graph to the right, and a negative C shifts it to the left.For y = sin(x - π/3), the phase shift is π/3. The graph i...

Standard Form of a Sine Function y = A \sin(B(x - C)) + D This is the general form used to graph transformations. 'A' controls the amplitude and reflection over the x-axis. 'B' controls the period. 'C' controls the phase (horizontal) shift. 'D' controls the vertical shift and midline. Period Formula \text{Period} = \frac{2\pi}{|B|} Use this formula to calculate the length of one full cycle of the sine wave. The standard period of y = sin(x) is 2π because B=1. Key Points for One Cycle Start: x = C \newline Quarter-point: x = C + \frac{\text{Period}}{4} \newline Mid-point: x = C + \frac{\text{Period}}{2} \newline Three-quarter-point: x = C + \frac{3 \cdot \text{Period}}{4} \newline End: x = C + \text{Period} These five x-values c...