Unit 7 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.

Lesson 11

Learning Focus

Solve quadratic equations graphically and algebraically.

Make connections between solving quadratic equations and graphing quadratic functions.

Lesson Summary

In this lesson, we learned methods for solving quadratic equations. Some quadratic equations can be solved using inverse operations and taking the square root of both sides of the equations. Some quadratic equations can be solved by factoring and using the zero product property. Some quadratic equations can be solved by completing the square and then using inverse operations. Quadratic equations that have real solutions can also be solved by graphing, and each of these algebraic methods has connections to graphing.

Lesson 12

Learning Focus

Understand and use a formula for solving quadratic equations.

Lesson Summary

In this lesson, we found a formula by completing the square for solving quadratic equations. We used the quadratic formula to find exact and approximate solutions to quadratic equations and connected those solutions to the graphs of quadratic functions. We learned to rewrite exact solutions so that fractions do not contain common factors in the numerator and denominator and the square roots do not contain factors that are perfect squares.

Lesson 13

Learning Focus

Solve quadratic equations efficiently and accurately.

Solve systems of quadratic and linear equations.

Lesson Summary

In this lesson, we compared methods for solving quadratic equations. We found that some equations lend themselves to one method, and other equations are more efficiently solved with other methods. Using technology to graph is always a useful way to check solutions.

Lesson 14

Learning Focus

Solve quadratic inequalities both graphically and algebraically.

Interpret solutions to quadratic inequalities that arise from context.

Solve a system of equations that contains both a quadratic and linear equation.

Lesson Summary

In this lesson, we developed a strategy for solving quadratic inequalities. The procedure involves solving the related quadratic equation and then using the graph or testing values to find the intervals that are solutions to the inequality. If the inequality represents a real context, the solutions must be interpreted so that they fit the situation.