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 identified common features of quadratic functions such as the vertex, line of symmetry, and the shape of the graph, which is called a parabola. We also learned how to efficiently find the domain and range of a quadratic function and determine if the graph of the function opens upward or downward.

Lesson 2

Learning Focus

Add and subtract linear and quadratic functions algebraically.

Add and subtract linear and quadratic functions graphically.

Lesson Summary

In this lesson, we learned to add and subtract polynomials, such as quadratic and linear functions. We learned that the procedure used for adding and subtracting is analogous to adding whole numbers because polynomials have the same structure as whole numbers. Polynomials are added by adding like terms. When subtracting polynomials, we can avoid sign errors by adding the opposite of each term.

Lesson 3

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 4

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 5

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 6

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 that the $x$ -intercepts are solutions to the equation $f (x) = 0$ , which allows us to use factoring as a method for solving a quadratic equation.

Lesson 7

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. Quadratic equations that have real solutions can also be solved by graphing, and each of these algebraic methods has connections to graphing.

Lesson 8

Learning Focus

Solve quadratic equations efficiently and accurately.

Identify information about the graph of a quadratic function from the equation.

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.