Write equations of cosine functions using properties

Learning Objectives Identify the amplitude, period, phase shift, and vertical shift from a graph or description of a sinusoidal function. Calculate the parameters 'a', 'b', 'c', and 'd' for the general cosine equation. Write the equation of a cosine function in the form y = a cos(b(x - c)) + d given its graph. Construct a cosine function's equation from key properties such as maximum/minimum values, period, and phase shift. Determine an appropriate phase shift for a cosine function by identifying the x-coordinate of a maximum point. Model real-world periodic phenomena by creating a corresponding cosine equation. Ever wondered how engineers model the alternating current in your home's outlets or how scientists predict ocean ti...

TermDefinitionExample Amplitude (a)The distance from the function's midline to its maximum or minimum value. It represents half the total vertical distance between the peak and trough and is always a non-negative value.For a wave that goes from a minimum of y=2 to a maximum of y=10, the amplitude is (10 - 2) / 2 = 4. Midline (Vertical Shift, d)The horizontal line that runs exactly halfway between the function's maximum and minimum values. Its equation is y = d.For a wave with a maximum of y=10 and a minimum of y=2, the midline is y = (10 + 2) / 2 = 6. So, d = 6. PeriodThe length of the smallest horizontal interval over which the function completes one full cycle.If a cosine wave starts at a peak at x=1 and reaches the next peak at x=5, its period is 5 - 1 = 4. The 'b'...

The General Cosine Function y = a \cos(b(x - c)) + d This is the standard form used to write the equation of any cosine function. 'a' is the amplitude, 'b' relates to the period, 'c' is the phase shift, and 'd' is the vertical shift (midline). Calculating Amplitude and Vertical Shift a = \frac{\text{Max} - \text{Min}}{2} \quad \text{and} \quad d = \frac{\text{Max} + \text{Min}}{2} Use these formulas when you are given the maximum and minimum y-values of the function. They allow you to quickly find the amplitude 'a' and the midline 'd'. Calculating the 'b' Value from the Period \text{Period} = \frac{2\pi}{|b|} \quad \implies \quad |b| = \frac{2\pi}{\text{Period}} This formula connects the visual len...