# Lesson 11

Finding Intersections

### Lesson Narrative

The purpose of this lesson is for students to solve systems of equations involving quadratics. Unlike systems of linear equations, systems involving quadratic functions can have 0, 1, or 2 distinct solutions, in addition to the infinitely many solutions case. Students begin the lesson by focusing on structure to identify possible solutions to a series of systems of equations (MP7). Building on this, students then combine their skills solving systems of equations using substitution and solving quadratic equations to identify all solutions to systems of equations involving at least one quadratic function.

The last activity is meant to make clear that while algebraic solving methods are useful, technology helps us identify solutions to systems of equations in which the steps needed to solve by hand are less obvious. This activity also asks students to consider what a possible factor of a polynomial expression written in standard form could be, which will be the focus of the following lessons leading into the Remainder Theorem.

### Learning Goals

Teacher Facing

• Calculate the solution to a system of polynomial equations.

### Student Facing

• Let’s think about two polynomials at once.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can find where two polynomial functions intersect.

Building On

Building Towards

### Glossary Entries

• end behavior

How the outputs of a function change as we look at input values further and further from 0.

This function shows different end behavior in the positive and negative directions. In the positive direction the values get larger and larger. In the negative direction the values get closer and closer to -3.

• multiplicity

The power to which a factor occurs in the factored form of a polynomial. For example, in the polynomial $$(x-1)^2(x+3)$$, the factor $$x-1$$ has multiplicity 2 and the factor $$x+3$$ has multiplicity 1.

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