This lesson is meant to be a culmination of what students have learned in the unit so far. In order to sketch a graph of a polynomial function from a factored equation, students must understand the relationship between the factors, zeros, and horizontal intercepts. They need to calculate the leading term of the equation in order to use the degree and the sign of the leading coefficient to identify the end behavior of the function.

A new idea in this lesson is the relationship between the multiplicity of a factor, which is the power to which the factor occurs in the factored form of a polynomial, and the shape of the graph at the intercept associated with that factor. While students previewed this idea in their study of quadratics in previous courses, here we name the effect and study the differences between factors with a multiplicity of 1, 2, and 3.

Students will have additional opportunities to sketch polynomials from factored equations in future lessons, but this lesson is the main opportunity to make solid connections between the shape of a graph and the structure of the factored equation for polynomial functions (MP7).