Unit 5 Quadratic Functions and Transformations

Lesson 1

Learning Focus

Model patterns with functions.

Compare and contrast linear and quadratic functions.

Lesson Summary

In this lesson, we modeled a quadratic and a linear function and compared representations. We combined a linear function and a quadratic function to examine the representations of the combined function. We noticed that when two functions are added, their corresponding outputs are added to make the new function.

Lesson 2

Learning Focus

Model a story context with a table, graph, and equation.

Identify features of a function from a graph.

Lesson Summary

In this lesson, we examined a quadratic function that was a model for area but had many features that were different than those we have seen previously. We learned that all quadratic functions have a linear rate of change and constant second difference, but some may be continuous and have intervals of increase and decrease depending on the domain.

Lesson 3

Learning Focus

Use patterns to efficiently graph quadratic functions from factored form.

Lesson Summary

In this lesson, we reviewed how to graph quadratic functions in factored form and standard form. 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 4

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 5

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 6

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 7

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 8

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 9

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.