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

Students are given the graph of f(x) = x² and are asked to compare the graphs of five other quadratic functions to the graph of f.

Type: Formative Assessment

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.

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

This lesson is intended to help you assess how well students are able to:

• Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions.
• Interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.

Type: Lesson Plan

This lesson unit is intended to help you assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials as well as recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x).

Type: Lesson Plan

This lesson is designed to be part of a sequence of lessons. It follows resource 52910 "BIOSCOPES Summer Institute 2013 - Mechanical Energy" and precedes resource 52705"BIOSCOPES Summer Institute 2013 - States of Matter." This lesson uses a predict, observe, and explain approach along with inquiry based activities to enhance student understanding of thermal energy and specific heat.

Type: Lesson Plan

This lesson plan includes transformations, domain, range, and symmetry for 5 different types of functions. Students will learn how to look at a function and determine what the graph will look like without plotting any points. Students will also be able to analyze the symmetry of functions algebraically.

Type: Lesson Plan

In this lesson, students will investigate the changes to the graph of a quadratic function when the function is modified in four different ways by inclusion of an additive or multiplicative constant. Students will work in groups to graph quadratic functions, prepare a display of their functions, and determine how the modification affects the graph of the quadratic function. Then, students participate in a gallery walk, where members of each group will share their findings with a small group of students. At the end, there is a class discussion to see if everyone had similar findings and to solidify the knowledge of translating quadratic functions.

Type: Lesson Plan

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 -1f(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

This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.

Type: Problem-Solving Task

The purpose of this task is to construct and use inverse functions to model a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point.

Type: Problem-Solving Task

Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation.

Type: Problem-Solving Task

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

The purpose of this task is to help students see the "why" behind properties of logs that are familiar but often just memorized (and quickly forgotten or misremembered). The task focuses on the verbal definition of the log, helping students to concentrate on understanding that a logarithm is an exponent, as opposed to completing a more computational approach.

Type: Problem-Solving Task

In this task, students explore the inverse relationship between an exponential function and a logarithmic function. The task is to determine the relevant composite functions, their graphs, and the domain and range of each.

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

Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions).

Type: Problem-Solving Task

In this task, students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible.

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 illustrates several components of standard MAFS.912.F-BF.2.4.c: Find Inverse Functions. Here, instead of presenting two functions and asking the students to decide which on is invertible, students are asked to complete a table of input-output pairs for the functions in such a way that one of the functions is invertible and the other one is not.

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

This 4 minute video gives step by step instruction of solving logarithmic equations.

Type: Tutorial

Type: Unit/Lesson Sequence

"Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Type: Unit/Lesson Sequence

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