Question

Determine an equation of the form y = a cos bx or y = a sin bx, where b \u003e 0, for the given graph. See Example 6.

Accepted Answer

Identify the key characteristics of the given graph: amplitude, period, phase shift, and vertical shift. These will help determine the values of $$a$$, $$b$$, and whether the function is sine or cosine. Determine the amplitude $$a by $$finding the distance from the midline of the graph to its maximum or minimum value. This is the absolute value of $$a in $$the equation $$y = a \sin(bx) or$$ $$y = a \cos(bx). $$Find the period of the function by measuring the length of one complete cycle on the x-axis. Use the formula for the period of sine and cosine functions: $$	ext{Period} = \frac{2\pi}{b}. $$Solve for $$b as$$ $$b = \frac{2\pi}{	ext{Period}}. $$Decide whether the function is sine or cosine by analyzing the starting point of the graph. If the graph starts at a maximum or minimum value at $$x=0$$, it is likely a cosine function. If it starts at the midline going upward or downward, it is likely a sine function. Write the equation in the form $$y = a \sin(bx) or$$ $$y = a \cos(bx)$$ using the values of $$a$$ and $$b$$ found, and include any phase shifts or vertical shifts if present.

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.
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Key Concepts

General Form of Sine and Cosine Functions

Amplitude and Period of Trigonometric Functions

Phase Shift and Choosing Between Sine and Cosine

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.<IMAGE>