*Include recognizing even and odd functions from their graphs and algebraic expressions for them.*

**Subject Area:**Mathematics

**Grade:**912

**Domain-Subdomain:**Functions: Building Functions

**Cluster:**Level 2: Basic Application of Skills & Concepts

**Cluster:**Build new functions from existing functions. (Algebra 1 - Additional Cluster) (Algebra 2 - Additional Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

**Date Adopted or Revised:**02/14

**Date of Last Rating:**02/14

**Status:**State Board Approved

**Assessed:**Yes

**Assessment Limits :**

Functions represented algebraically are limited to linear, quadratic, or

exponential.Functions represented using tables or graphs are not limited to linear,

quadratic, or exponential.Functions may be represented using tables or graphs.

Functions may have closed domains.

Functions may be discontinuous.

Items should have a single transformation.

**Calculator :**Neutral

**Clarification :**

Students will determine the value of k when given a graph of the

function and its transformation.Students will identify differences and similarities between a function

and its transformation.Students will identify a graph of a function given a graph or a table of

a transformation and the type of transformation that is represented.Students will graph by applying a given transformation to a function.

Students will identify ordered pairs of a transformed graph.

Students will complete a table for a transformed function.

**Stimulus Attributes :**Items should be given in a mathematical context.

Items must use function notation.

Items may present a function using an equation, a table of values, or

a graph.**Response Attributes :**

Items may require the student to explain or justify a transformation

that has been applied to a function.Items may require the student to explain how a graph is affected by a

value of k.Items may require the student to find the value of k.

Items may require the student to complete a table of values.

**Test Item #:**Sample Item 1**Question:**The table below shows the values for the function y = f(x).

Complete the table for the function

**Difficulty:**N/A**Type:**TI: Table Item

## Related Courses

## Related Access Points

## Related Resources

## Assessment

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Problem-Solving Tasks

## Unit/Lesson Sequence

## Virtual Manipulatives

## MFAS Formative Assessments

Students are asked to use technology to graph exponential functions and then to describe the effect on the graph of changing the parameters of the function.

Students are asked to compare the graphs of four different linear functions to the graph of *f*(*x*) = *x*.

Students are given the graph of *f*(*x*) = *x*^{2} and are asked to compare the graphs of five other quadratic functions to the graph of *f*.

Students are given the graphs of three absolute values functions and are asked to write the equation of each.

## Original Student Tutorials Mathematics - Grades 9-12

Visualize the effect of using a value of k in both *kf*(*x*) or *f*(*kx*) when k is greater than zero in this interactive tutorial.

Learn how reflections of a function are created and tied to the value of *k* in the mapping of *f*(*x*) to -1*f*(*x*) in this interactive tutorial.

Explore translations of functions on a graph that are caused by *k* in this interactive tutorial. GeoGebra and interactive practice items are used to investigate linear, quadratic, and exponential functions and their graphs, and the effect of a translation on a table of values.

## Student Resources

## Original Student Tutorials

Visualize the effect of using a value of k in both *kf*(*x*) or *f*(*kx*) when k is greater than zero in this interactive tutorial.

Type: Original Student Tutorial

Learn how reflections of a function are created and tied to the value of *k* in the mapping of *f*(*x*) to -1*f*(*x*) in this interactive tutorial.

Type: Original Student Tutorial

Explore translations of functions on a graph that are caused by *k* in this interactive tutorial. GeoGebra and interactive practice items are used to investigate linear, quadratic, and exponential functions and their graphs, and the effect of a translation on a table of values.

Type: Original Student Tutorial

## Problem-Solving Tasks

This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.

Type: Problem-Solving Task

This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.

Type: Problem-Solving Task

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Type: Problem-Solving Task

This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function *f*. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."

Type: Problem-Solving Task

This task asks students to determine whether a the set of given functions is odd, even, or neither.

Type: Problem-Solving Task

## Virtual Manipulatives

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This is a graphing tool/activity for students to deepen their understanding of polynomial functions and their corresponding graphs. This tool is to be used in conjunction with a full lesson on graphing polynomial functions; it can be used either before an in depth lesson to prompt students to make inferences and connections between the coefficients in polynomial functions and their corresponding graphs, or as a practice tool after a lesson in graphing the polynomial functions.

Type: Virtual Manipulative

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

## Parent Resources

## Problem-Solving Tasks

This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.

Type: Problem-Solving Task

This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.

Type: Problem-Solving Task

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Type: Problem-Solving Task

This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function *f*. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."

Type: Problem-Solving Task

This task asks students to determine whether a the set of given functions is odd, even, or neither.

Type: Problem-Solving Task

## Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative