# Standard 1: Understand, compare and analyze properties of functions.

General Information
Number: MA.912.F.1
Title: Understand, compare and analyze properties of functions.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Functions

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.912.F.1.AP.1a
Given an equation or graph that defines a function, identify the function type as either linear, quadratic, or exponential.
MA.912.F.1.AP.1b
Given an input-output table with an accompanying graph, determine a function type, either linear, quadratic, or exponential that could represent it.
MA.912.F.1.AP.2
Given a function represented in function notation, evaluate the function for an input in its domain.
MA.912.F.1.AP.3
Given a real-world situation represented graphically or algebraically, identify the rate of change as positive, negative, zero or undefined.
MA.912.F.1.AP.5
Identify key features of linear and quadratic functions each represented in the same way algebraically or graphically (key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; end behavior).
MA.912.F.1.AP.6
Identify key features of linear, quadratic or exponential functions each represented in a different way algebraically or graphically (key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; end behavior).
MA.912.F.1.AP.7
Compare key features of two functions each represented algebraically or graphically.
MA.912.F.1.AP.8
Select whether a linear or quadratic function best models a given real-world situation.
MA.912.F.1.AP.9
Select whether a function is even, odd or neither when represented algebraically.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

Students are given a linear and an exponential function, one represented verbally and the other by a table. Then students are asked to compare the rate of change in each in the context of the problem.

Type: Formative Assessment

Comparing Linear and Exponential Functions:

Students are given a linear function represented by an equation and an exponential function represented by a graph in a real-world context and are asked to compare the rates of change of the two functions.

Type: Formative Assessment

Comparing Linear Functions:

Students are given two linear functions, one represented by a graph and the other by an equation, and asked to compare their intercepts in the context of a problem.

Type: Formative Assessment

Recursive Sequences:

Students are asked to find the first five terms of a sequence defined recursively, explain why the sequence is a function, and describe its domain

Type: Formative Assessment

Prove Linear:

Students are asked to prove that a linear function grows by equal differences over equal intervals.

Type: Formative Assessment

Prove Exponential:

Students are asked to prove that an exponential function grows by equal factors over equal intervals.

Type: Formative Assessment

Linear or Exponential?:

Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.

Type: Formative Assessment

Students are asked to compare a quadratic and an exponential function in context.

Type: Formative Assessment

Fit a Function:

Students are given a set of data and are asked to use technology to create a scatter plot and write a function that fits the data set.

Type: Formative Assessment

Cell Phone Battery Life:

Students are asked to interpret statements that use function notation in the context of a problem.

Type: Formative Assessment

Nonlinear Functions:

Students are asked to provide an example of a nonlinear function and explain why it is nonlinear.

Type: Formative Assessment

Compare Linear and Exponential Functions:

Students are asked to compare a linear function and an exponential function in context.

Type: Formative Assessment

What Is the Value?:

Students are asked to determine the corresponding input value for a given output using a table of values representing a function, f.

Type: Formative Assessment

Pizza Palace:

Students are given a table of functional values in context and are asked to find the average rate of change over a specific interval.

Type: Formative Assessment

Air Cannon:

Students are given a graph of an exponential function and are asked to calculate and compare the average rate of change over two different intervals of time.

Type: Formative Assessment

Estimating the Average Rate of Change:

Students are asked to estimate the average rate of change of a nonlinear function over two different intervals given its graph.

Type: Formative Assessment

Identifying Rate of Change:

Students are asked to calculate and interpret the rate of change of a linear function given its graph.

Type: Formative Assessment

What Is the Function Notation?:

Students are asked to use function notation to rewrite the formula for the volume of a cube and to explain the meaning of the notation.

Type: Formative Assessment

Graphs and Functions:

Students are asked to determine the value of a function, at an input given using function notation, by inspecting its graph.

Type: Formative Assessment

Evaluating a Function:

Students are asked to evaluate a function at a given value of the independent variable.

Type: Formative Assessment

## Lesson Plans

Stop That Arguing:

Students will explore representing the movement of objects and the relationship between the various forms of representation: verbal descriptions, value tables, graphs, and equations. These representations include speed, starting position, and direction. This exploration includes brief direct instruction, guided practice in the form of a game, and independent practice in the form of a word problem. Students will demonstrate understanding of this concept through a written commitment of their answer to the word problem supported with evidence from value tables, graphs, and equations.

Type: Lesson Plan

Select a Healthcare Plan:

Students are asked to determine a procedure for ranking healthcare plans based on their assumptions and the cost of each plan given as a function. Then, they are asked to revise their ranking based on a new set of data.

Type: Lesson Plan

Appreciation for Car Depreciation:

Students will use information from the internet or a car dealership's advertisement to identify a car and determine the future value of the car using different depreciation rates over different intervals of time. Students will graph their data to show exponential decay and compare to a linear decrease on the same graph.

Type: Lesson Plan

Whose Line Is It Anyway?:

In this lesson, students will use graphing calculators to explore linear equations in the form y = mx + b. They will observe the graphs of equations with different values of slope and y-intercept. They will draw conclusions about how the value of slope and y-intercept are visible in the appearance of the graph.

Type: Lesson Plan

How Fast Do Objects Fall?:

Students will investigate falling objects with very low air friction.

Type: Lesson Plan

Which Function?:

This activity has students apply their knowledge to distinguish between numerical data that can be modeled in linear or exponential forms. Students will create mathematical models (graph, equation) that represent the data and compare these models in terms of the information they show and their limitations. Students will use the models to compute additional information to predict future outcomes and make conjectures based on these predictions.

Type: Lesson Plan

How much is your time worth?:

This lesson is designed to help students solve real-world problems involving compound and continuously compounded interest. Students will also be required to translate word problems into function models, evaluate functions for inputs in their domains, and interpret outputs in context.

Type: Lesson Plan

Graphing vs. Substitution. Which would you choose?:

Students will solve multiple systems of equations using two methods: graphing and substitution. This will help students to make a connection between the two methods and realize that they will indeed get the same solution graphically and algebraically.  Students will compare the two methods and think about ways to decide which method to use for a particular problem. This lesson connects prior instruction on solving systems of equations graphically with using algebraic methods to solve systems of equations.

Type: Lesson Plan

Freeze:

In this lesson students will learn how to write equations in function notation when given a real-world scenario. Students will work in groups to determine an equation for a given scenario, as well as, write a scenario for a given equation.

Type: Lesson Plan

Dancing Polynomials/Graph Me Baby:

Dancing Polynomials is designed to lead students from the understanding that the equation of a line produces a linear pattern to the realization that using an exponent greater than one will produce curvature in a graph and that further patterns emerge allowing students to predict what happens at the end of the graph. Using graphing calculators, students will examine the patterns that emerge to predict the end behavior of polynomial functions. They will experiment by manipulating equations superimposed onto landmarks in the shape of parabolas and polynomial functions. An end behavior song and dance, called "Graph Me Baby" will allow students to become graphs to physically understand the end behavior of the graph.

Type: Lesson Plan

This lesson introduces students to the quadratic parent function, as well as reinforces some key features of quadratic functions. It allows students to explore basic transformations of quadratic functions and provides a note-taking sheet for students to organize their learning. There is a "FUN" cut and paste activity for students to match quadratic graphs with verbal descriptions and their equations.

Type: Lesson Plan

This lesson covers quadratic translations as they relate to vertex form of a quadratic equation. Students will predict what will happen to the graph of a quadratic function when more than one constant is in a quadratic equation. Then, the students will graph quadratic equations in vertex form using their knowledge of the translations of a quadratic function, as well as describe the translations that occur. Students will also identify the parent function of any quadratic function as $f\left(x\right)={x}^{2}$.

Type: Lesson Plan

My Candles are MELTING!:

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph each to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Turning Tires Model Eliciting Activity:

The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

Type: Lesson Plan

## Original Student Tutorials

Functions, Functions, Everywhere: Part 2:

Continue exploring how to determine if a relation is a function using graphs and story situations in this interactive tutorial.

This is the second tutorial in a 2-part series. Click HERE to open Part 1.

Type: Original Student Tutorial

Travel with Functions:

Learn how to evaluate and interpret function notation by following Melissa and Jose on their travels in this interactive tutorial.

Type: Original Student Tutorial

Changing Rates:

Learn how to calculate and interpret an average rate of change over a specific interval on a graph in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Experts

Birdsong Series: Mathematically Modeling Birdsong:

Richard Bertram discusses his mathematical modeling contribution to the Birdsong project that helps the progress of neuron and ion channel research.

Type: Perspectives Video: Expert

Birdsong Series: STEM Team Collaboration :

Researchers Frank Johnson, Richard Bertram, Wei Wu, and Rick Hyson explore the necessity of scientific and mathematical collaboration in modern neuroscience, as it relates to their NSF research on birdsong.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Hurricane Dennis & Failed Math Models:

What happens when math models go wrong in forecasting hurricanes?

Type: Perspectives Video: Professional/Enthusiast

Graphing Torque and Horsepower for Dyno-mite Racing:

SCCA race car drivers discuss how using a chassis dyno to graph horsepower and torque curves helps them maximize potential in their race cars.

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Kites, Wind, and Speed:

Lofty ideas about kites helped power a kayak from California to Hawaii.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Water Usage Rates:

A seafaring teacher filters all the good information you need to understand water purification rates for distance traveling.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

Programming Mathematics: Algebra, and Variables to control Open-source Hardware:

If you are having trouble understanding variables, this video might help you see the light.

Type: Perspectives Video: Teaching Idea

## Text Resource

By the Skin of Their Suits:

This informational text resource is intended to support reading in the content area. The text discusses the two main factors that control the speed of a competitive swimmer: power and drag. The reader is then presented with mathematical formulas that determine these factors. The text also discusses the technological advances that have come about in the swimsuit industry. The text even entertains the idea of "technological doping" and allows the reader to question whether advanced swimsuits are hurting the competitiveness of swimming.

Type: Text Resource

## Virtual Manipulative

Functions and Vertical Line Test:

This lesson is designed to introduce students to the vertical line test for functions as well as practice plotting points and drawing simple functions. The lesson provides links to discussions and activities related to the vertical line test and functions as well as suggested ways to integrate them into the lesson.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Functions, Functions, Everywhere: Part 2:

Continue exploring how to determine if a relation is a function using graphs and story situations in this interactive tutorial.

This is the second tutorial in a 2-part series. Click HERE to open Part 1.

Type: Original Student Tutorial

Travel with Functions:

Learn how to evaluate and interpret function notation by following Melissa and Jose on their travels in this interactive tutorial.

Type: Original Student Tutorial

Changing Rates:

Learn how to calculate and interpret an average rate of change over a specific interval on a graph in this interactive tutorial.

Type: Original Student Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.