Write equations of sine functions from graphs

Learning Objectives Identify the amplitude, period, phase shift, and vertical shift of a sinusoidal function from its graph. Calculate the parameters a, b, h, and k for the standard sine equation y = a sin(b(x - h)) + k. Determine the correct sign of the amplitude 'a' based on the graph's behavior at the phase shift. Synthesize the calculated parameters into a complete and accurate sine function equation. Write multiple equivalent sine equations for a single graph by choosing different phase shifts. Analyze graphs with various scales on the x-axis (e.g., multiples of π) to determine the period and phase shift. Ever wondered how sound engineers model a sound wave or how astronomers predict tides just by looking at a graph? 🌊 Let's learn how to translate t...

TermDefinitionExample Midline (Vertical Shift)The horizontal line that passes exactly halfway between the function's maximum and minimum values. Its equation is y = k, where k is the vertical shift.If a sine wave oscillates between a maximum of y=5 and a minimum of y=-1, its midline is y = (5 + (-1))/2 = 2. Amplitude (a)The distance from the midline to either the maximum or minimum value of the function. It represents half the total vertical distance between the max and min points and is always positive.For a wave oscillating between y=5 and y=-1, the amplitude is (5 - (-1))/2 = 3. PeriodThe length of the shortest horizontal interval over which the function's values repeat. It is the length of one full cycle.If a sine wave starts a cycle at x=0 and completes it at x=4π, its peri...

Standard Sine Function Equation y = a \sin(b(x - h)) + k This is the standard form used to represent a transformed sine function. 'a' controls amplitude and reflection, 'b' controls the period, 'h' is the phase (horizontal) shift, and 'k' is the vertical shift (midline). Parameter Calculation Formulas k = \frac{Max + Min}{2} \quad |a| = \frac{Max - Min}{2} \quad b = \frac{2\pi}{Period} Use these formulas to find the values of k, a, and b directly from the maximum value, minimum value, and period identified from the graph. The phase shift 'h' is found by direct observation of a starting point.