Experience First

Students will use Desmos to explore the relationship between the roots, factors, y-intercepts, and turning points of a polynomial.  Have them graph each function and answer the corresponding questions.  Make sure they really pay attention to the behavior around the x-intercepts.  Does the curve go straight through?  Does it “bounce” and change directions?  How do the factors play into this?  We want them to see the repeated factors have different behavior at the x-intercepts to introduce the concept of multiplicity. 

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Formalize Later

When you formalize, make sure you use the proper language to define x- and y-intercepts.  The x-intercept isn’t just the “opposite sign of the factor” like we anticipate many students will put in 1d, but the solution when you set function equal to 0.  The y-intercept isn’t just where the graph crosses the y-axis, but the value of y when you evaluate f(0).

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Show on the graph where the multiplicity changes the behavior by outlining the “bounce” versus the “goes through.”  Connect the word x-intercept to root and zero since we will be using these interchangeably moving forward.

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Next lesson we will explore polynomials in the long run to create a complete picture of how polynomial graphs are related to their equations.