This problem shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large.

The purpose of this task is twofold: first using technology to study the behavior of some exponential and logarithmic graphs and secondly to manipulate some explicit logarithmic and exponential expressions. Although not asked in the task body, the teacher may wish to prompt students to explain why the two graphs behave as they do as the base b varies: that is, a larger value of b between 1 and 2 makes the exponential graph grow faster and the logarithmic graph grow more slowly as x increases.

The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. In general, an exponential function f(t)=ab^t has two parameters. The parameter a is interpreted as the starting value (when t represents time), and b represents the growth rate -- the amount the quantity is multiplied by each time the value of t is incremented by 1.

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.

This task and its companion, F-BF Exponentials and Logarithms I, is designed to help students gain facility with properties of exponential and logarithm functions resulting from the fact that they are inverses.

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t=0.

The goal of this task is to develop an understanding of why rational exponents are defined as they are (N-RN.1), however it also raises important issues about distinguishing between linear and exponential behavior (F-LE.1c) and it requires students to create an equation to model a context (A-CED.2

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The table below shows some input-output pairs of two functions $f$ and $g$ that agree for the values that are given but some of their output values are...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Standard maps of the earth are broken into a grid of latitude lines (east-west) and longitude lines (north-south). Consider the function, $N(\ell)$, th...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: A model airplane pilot is practicing flying her airplane in a big loop for an upcoming competition. At time $t=0$ her airplane is at the bottom of the ...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: You are a marine biologist working for the Environmental Protection Agency (EPA). You are concerned that the rare coral mathemafish population is being...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The table below shows historical estimates for the population of London. Year18011821 18411861 18811901 1921 1939 1961 London population 1,100,000 1,60...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Pictured below are the graphs of four different functions, defined in terms of eight constants: $a, b, c, k, m, p, q, \text{ and } r.$ The equations of...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: In this task we are going to investigate the graphs of $\displaystyle{f(x) = \frac{1}{x+a}}$ and $\displaystyle{g(x) = \frac{1}{x^2+b}}.$ Move the slid...

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.