Write equations of cosine functions from graphs

Learning Objectives Identify the amplitude, period, phase shift, and vertical shift of a cosine function from its graph. Calculate the 'b' value for the cosine equation using the period found from the graph. Determine the vertical shift (k) by finding the midline of the function. Determine the amplitude (a) and its sign (positive or negative) based on the graph's orientation. Identify the phase shift (h) by locating a maximum point on the graph. Synthesize the identified parameters (a, b, h, k) to construct a complete and accurate cosine equation. Write multiple equivalent cosine equations for a single graph by choosing different phase shifts. Ever wondered how engineers model the cyclical motion of a piston or how scientists predict ocean tides? 🌊 They tra...

TermDefinitionExample Midline (Vertical Shift, k)The horizontal line that passes exactly halfway between the function's maximum and minimum values. The y-value of this line is the vertical shift, 'k'.If a function's maximum is y=8 and its minimum is y=2, the midline is y = (8+2)/2 = 5. So, k=5. Amplitude (a)The distance from the midline to either the maximum or minimum value of the function. It represents the 'height' of the wave. The value of 'a' is positive if the function reaches a maximum at its starting point (after the phase shift) and negative if it reaches a minimum.For a function with a maximum of 8 and a minimum of 2, the amplitude is |a| = (8-2)/2 = 3. Period (P)The length of the smallest horizontal interval over which the function's...

The Standard Cosine Equation y = a \cos(b(x - h)) + k This is the target form for our equation. 'a' is the amplitude and reflection, 'b' relates to the period, 'h' is the phase (horizontal) shift, and 'k' is the vertical shift (midline). Midline and Amplitude Formulas k = \frac{y_{max} + y_{min}}{2} \quad \text{and} \quad |a| = \frac{y_{max} - y_{min}}{2} Use the maximum and minimum y-values from the graph to directly calculate the vertical shift (k) and the absolute value of the amplitude (a). Period and 'b' Value Formula P = \frac{2\pi}{|b|} \quad \Longleftrightarrow \quad |b| = \frac{2\pi}{P} After measuring the period (P) from the graph (the length of one full cycle), use this formula to calculate the value of &#0...