# Modeling Riverbeds with Polynomials

### Modeling Riverbeds with Polynomials

Student Outcomes

• Students learn to fit polynomial functions to data values.

### New York State Common Core Math Algebra II, Module 1, Lesson 20

Worksheets for Algebra II, Module 1, Lesson 20

Classwork

Mathematical Modeling Exercise The Environmental Protection Agency (EPA) is studying the flow of a river in order to establish flood zones. The EPA hired a surveying company to determine the flow rate of the river, measured as volume of water per minute. The firm set up a coordinate system and found the depths of the river at five locations as shown on the graph below. After studying the data, the firm decided to model the riverbed with a polynomial function and divide the cross-sectional area into six regions that are either trapezoidal or triangular so that the overall area can be easily estimated. The firm needs to approximate the depth of the river at two more data points in order to do this.

Draw the four trapezoids and two triangles that will be used to estimate the cross-sectional area of the riverbed.

Example 1

Find a polynomial 𝑃 such that 𝑃(0) = 28, 𝑃(2) = 0, and 𝑃(8) = 12.

Example 2

Find a degree 3 polynomial 𝑃 such that 𝑃(−1) = −3, 𝑃(0) = −2, 𝑃(1) = −1, and 𝑃(2) = 6

Lesson Summary

A linear polynomial is determined by 2 points on its graph.
A degree 2 polynomial is determined by 3 points on its graph.
A degree 3 polynomial is determined by 4 points on its graph.
A degree 4 polynomial is determined by 5 points on its graph.
The remainder theorem can be used to find a polynomial 𝑃 whose graph will pass through a given set of points.

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