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Sara's roller coaster needs to be continuous. That means that the y-values at the boundary points (x = 0 and x = 2π) need to agree from the left and the right.

Determine the y-value of the roller coaster at x = 0: −2cos(0) + 3 = 1

Next, determine the y-value of the roller coaster at x = 2π: −cos(2(2π)) −4 = −5

We are ready to find the equation of the middle piece of the piecewise function. This piece has two unknowns, a and b. Fortunately, in Step 2 and Step 3, we found two values that exist on the middle equation, if the piecewise is continuous.
Write and solve a system for a and b:
1 = acos(bx) −2 when x = 0
−5 = acos(bx) −2 when x = 2π

Use the eTool below to dynamically adjust the graph.
Click the link at right for the full version of the eTool: Calc 2-23 HW eTool