Graph cosine functions

Learning Objectives Identify the amplitude, period, phase shift, and vertical shift from the equation y = A cos(B(x - C)) + D. Graph the parent function y = cos(x) and at least two transformations. Determine the five key points (maximum, midline intercepts, minimum) of one cycle of a cosine function. Derive the equation of a cosine function given its graph. Analyze the effect of each parameter (A, B, C, D) on the graph of the parent function. Apply the first derivative to find the slope of the tangent line at any point on a cosine curve. Ever wondered how noise-cancelling headphones work or how we model the ebb and flow of tides? 🌊 The secret lies in understanding and manipulating waves, starting with the cosine graph! This tutorial will guide you through graphing the cosi...

TermDefinitionExample AmplitudeThe amplitude is half the 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 y = 3 cos(x), the amplitude is |3| = 3. The graph oscillates between a maximum of 3 and a minimum of -3. PeriodThe period is the length of one complete cycle of the graph, measured along the x-axis. It is the horizontal distance after which the function's values begin to repeat.For y = cos(2x), the period is 2π / |2| = π. The graph completes one full wave over an interval of π units. Phase ShiftThe phase shift is the horizontal translation of the graph from its standard position. It is determined by the value of C in the form y = A cos(B(x - C)) + D.For y = cos(x - π/2),...

General Form of a Cosine Function y = A \cos(B(x - C)) + D This is the standard form used to analyze and graph transformations of the cosine function. A controls amplitude and reflection, B controls the period, C controls the phase shift, and D controls the vertical shift and midline. Period Calculation \text{Period} = \frac{2\pi}{|B|} Use this formula to calculate the length of one full cycle of the cosine wave. Remember that the standard period of y = cos(x) is 2π, so the B value compresses or stretches the graph horizontally. Key Points Interval \text{Interval} = \frac{\text{Period}}{4} To accurately sketch one cycle, divide the period by 4. This gives the horizontal distance between the five key points (maximum, midline, minimum, midline, maximum).