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Sara is designing a model that will represent the path of a roller coaster. She has determined the beginning and the end parts of the track, but needs to find a formula for the middle section that will join the other segments. She decided that she wants one peak in this middle section, not including its boundaries. Find values of and which will make her function, given below, continuous. To help you visualize this, use the .

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

Determine the -value of the roller coaster at :

Next, determine the -value of the roller coaster at :

We are ready to find the equation of the middle piece of the piecewise function. This piece has two unknowns, and . 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 and :

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