Unit 2 Structures of Quadratic Expressions

Lesson 1

Learning Focus

Find patterns in the equations and graphs of quadratic functions.

Lesson Summary

In this lesson, we explored transformations of the function $y = x^{2}$ . We found vertical and horizontal shifts, reflections, and vertical stretches of the parabola. We justified why changes to the equation transform the graph, using tables and our understanding of functions.

Lesson 2

Learning Focus

Write equations for functions that are transformations of $f (x) = x^{2}$ .

Find efficient methods for graphing transformations of $f (x) = x^{2}$ .

Lesson Summary

In this lesson, we learned to graph quadratic functions that have a combination of transformations. We found that the vertex form of the equation of a quadratic function makes it easy to find the vertex and identify the transformations. We wrote equations in vertex form from graphs and tables, using our understanding of transformations and the features of parabolas.

Lesson 3

Learning Focus

Find the square of a binomial expression.

Recognize a perfect square trinomial.

Create perfect squares from partial areas.

Find relationships between terms in a perfect square trinomial.

Lesson Summary

In this lesson, we connected area models for multiplication to show how to multiply binomials to get a perfect square trinomial. We learned to recognize a perfect square trinomial by looking for a relationship between the second and third terms. We also worked to create a perfect square when given the first two terms of a trinomial.

Lesson 4

Learning Focus

Find a process for completing the square that works on all quadratic functions.

Adapt diagrams to become more efficient in completing the square.

Lesson Summary

In this lesson, we solidified a process for completing the square with expressions in the form $a x^{2} + b x + c$ with $a \neq 1$ . We learned an algebraic procedure that goes along with an open diagram that supports our work. We also verified that the expression obtained by completing the square was equivalent to the original expression using the distributive property.

Lesson 5

Learning Focus

Use completing the square to change the form of a quadratic equation.

Graph quadratic equations given in standard form.

Lesson Summary

In this lesson, we learned to graph a quadratic function in standard form. We used the process of completing the square to help identify the transformations and locate the vertex. From there, we were able to use the quick-graph method to graph the parabola.

Lesson 6

Learning Focus

Multiply two binomials using diagrams.

Factor a trinomial using diagrams.

Lesson Summary

In this lesson, we used area model diagrams to multiply binomials and factor trinomials. We identified a relationship between the numbers in the factors and the numbers in the equivalent trinomial that helps us to find the factors more easily.

Lesson 7

Learning Focus

Find patterns in signs and numbers to help factor and multiply expressions.

Use area model diagrams to multiply binomials with different signs.

Use area model diagrams to factor trinomials when some of the terms are negative.

Lesson Summary

In this lesson, we learned to multiply binomials that had both positive and negative numbers in the factors. We found a useful pattern called “difference of squares” that occurs when the two factors have the same numbers but opposite signs. We learned to factor trinomials that have both positive and negative terms using sign and number patterns to be sure that the factored expression is equivalent to the trinomial.

Lesson 8

Learning Focus

Use diagrams to factor trinomial expressions when the leading coefficient is not $1$ .

Lesson Summary

In this lesson, we learned to factor trinomials in the form $a x^{2} + b x + c$ when $a \neq 1$ . Sometimes the terms have a common factor that can be factored out, leaving an expression that is much easier to work with. When there is not a common factor, diagrams can be used to help think about the number and sign combinations that work to make the factored expression equivalent to the trinomial.

Lesson 9

Learning Focus

Find patterns to efficiently graph quadratic functions from factored form.

Lesson Summary

In this lesson, we learned to use the factored form of a quadratic equation to graph parabolas. We learned to find the $x$ -intercepts from the factors, then find the line of symmetry between the $x$ -intercepts. Once we knew the line of symmetry, we could find the vertex. We observed several patterns that helped to make factored form an efficient way to graph quadratics.

Lesson 10

Learning Focus

Choose the most efficient form of a quadratic function.

Become efficient and accurate in converting from one quadratic form to another.

Become efficient and accurate in identifying features of the graph of quadratic functions from a given form.

Lesson Summary

In this lesson, we learned to make strategic choices about the most efficient form for working with the graph of a quadratic function. We considered which form is most efficient for obtaining features like the vertex, $x$ -intercepts, $y$ -intercept, the vertical stretch, and reflection. We also considered which form will be most efficient to convert from standard form, knowing that some trinomials do not factor easily and some trinomials make completing the square complicated.