# Grades 9/10/11 Quadratic Relations

Graphs and tables. Standard, factored, and vertex forms. Algebra of quadratic relations. Quadratic equations. Intersections of lines and parabolas.

This is one of seven strands of the CEMC Grade 9/10/11 courseware. The other strands and more information about this courseware is available on the Grade 9/10/11 homepage.

### Units

## Unit 1: Basic Properties of Quadratic Relations

This lesson introduces quadratic relations to students by examining tables of values and second differences.

This lesson extends our study of quadratic relations, with an emphasis on using second differences to determine unknown values in a table.

This lesson defines key terms related to parabolas (vertex, zeros, axis of symmetry, etc.) and interprets those terms within a given context.

Students will compare the features of the graphs of \(y=x^2\) and \(y=2^x\).

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 2: Algebraic Representations of Quadratic Relations

This lesson focuses on the algebraic representation of a quadratic relation in standard form, and how this representation relates to the second differences, graphs, and other algebraic representations of the relation.

In this lesson, students will investigate the factored form equation of a quadratic relation.

In this lesson, students will investigate the vertex form equation of a quadratic relation. We will use vertex form equations to solve a variety of problems and see how to convert a factored form equation to a vertex form equation.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 3: Algebraic Skills

In this lesson, students will apply the distributive property to multiply and square binomials. They will practise expanding and simplifying in the context of quadratic relations, particularly changing equations from factored or vertex form to standard form.

In this lesson, students will begin learning about factoring. Factoring is the algebraic process that converts a standard form equation into a factored form equation, making it easy to find the zeros. This lesson focuses on common factoring and factoring trinomials of the form \(ax^2+bx+c\).

In this lesson, students will complete their study of factoring. This lesson focuses on difference of squares, perfect squares, and combining multiple types of factoring together.

In this lesson, students will be introduced to the process of completing the square, for the purpose of changing the equation of a quadratic relation from standard form to vertex form. They will move from more simple examples to more complex cases with fractions, and apply the process in contexts where they need to find the vertex.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 4: Graphing Quadratic Relations

In this lesson, transformations will be explored. We will consider translations, reflections in the \(x\)-axis, and vertical stretches of parabolas. We will look at the relationship between these transformations and the equation of a parabola.

This lesson discusses the relationship between the graph of a quadratic relation and its equation in vertex form. Methods of generating the graph of a relation from its equation, including the application of transformations, will be explored, as well as how to identify the equation of a relation from its graph.

In this lesson, a brief recap of factored form is given. We then extend our understanding of factored form to graph the quadratic relation.

This lesson will discuss methods of sketching the graph of a quadratic relation whose equation is given in standard form, \(y=ax^2+bx+c\). These methods include writing the equation in vertex form first by completing the square, or factoring the equation if possible to find the zeros.

## Unit 5: Solving Problems Involving Quadratic Relations

In this lesson, students will be introduced to quadratic equations and begin to solve them by graphing, applying inverse operations, or factoring. They will also explore different methods of checking their solutions.

In this lesson, we develop a new way to find the zeros of a quadratic relation given standard form. The Quadratic Formula will be derived and then used in a variety of contexts.

In this lesson, we recall that a quadratic relation can have either 0, 1, or 2 zeros. We develop ways to predict how many zeros the relation will have by analyzing the equation of the relation.

In this lesson, students will explore the possible number of points of intersection between a linear relation and a quadratic relation. They will determine coordinates of the point(s) of intersection of a linear and quadratic relation both graphically and algebraically, and use the discriminant to determine how many times a linear and quadratic relation will intersect.

In this lesson, students will solve problems involving quadratic relations relating to various applications. These problems involve both solving quadratic equations and finding a maximum or minimum, and will require the algebraic methods discussed in previous lessons. The method of partial factoring will also be introduced for determining the vertex.