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 lesson unit is intended to help teachers assess how well students are able to: model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions; and 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.

In this task students have the opportunity to construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

The purpose of this problem is to have students compose functions using tables of values only.

There is a natural (and complicated!) predator-prey relationship between the fox and rabbit populations, since foxes thrive in the presence of rabbits, and rabbits thrive in the absence of foxes. However, this relationship, as shown in the given table of values, cannot possibly be used to present either population as a function of the other. This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations.

The example of rabbits and foxes was introduced in the task (8-F Foxes and Rabbits) to illustrate two functions of time given in a table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation.

The example of rabbits and foxes was introduced in 8-F Foxes and Rabbits to illustrate two functions of time given in a table. The same situation was used in F-TF Foxes and Rabbits 2 to find trigonometric functions modeling the data in the table. The previous situation was somewhat unrealistic since we were able to find functions that fit the data perfectly. In this task, on the other hand, we do some legitimate modelling, in that we come up with functions that approximate the data well, but do not perfectly match, the given data.

This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule.

This site teaches Building Functions to High Schoolers through a series of 1355 questions and interactive activities aligned to 13 Common Core mathematics skills.

This site teaches Interpreting Functions to High Schoolers through a series of 3578 questions and interactive activities aligned to 38 Common Core mathematics skills.

This site teaches Linear, Quadratic, and Exponential Models to High Schoolers through a series of 137 questions and interactive activities aligned to 6 Common Core mathematics skills.

This site teaches Trigonometric Functions to High Schoolers through a series of 915 questions and interactive activities aligned to 15 Common Core mathematics skills.

This lesson unit is intended to help teachers assess how well students are able to: articulate verbally the relationships between variables arising in everyday contexts; translate between everyday situations and sketch graphs of relationships between variables; interpret algebraic functions in terms of the contexts in which they arise; and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.

This site teaches Functions to 8th graders through a series of 1920 questions and interactive activities aligned to 17 Common Core mathematics skills.

This lab is an investigative lab to help students make a visual connection between quadratic and polynomial functions and their graphs. The lab uses desmos.com and takes about 30-45 minutes.

This task addresses an important issue about inverse functions. In this case the function f is the inverse of the function g but g is not the inverse of f unless the domain of f is restricted.

This task requires students to recognize the graphs of different (positive) powers of x. There are several important aspects to these graphs. First, the graphs of even powers of x all open upward as x grows in the positive or negative direction. The larger the even power, the flatter these graphs look near 0 and the more rapidly they increase once the distance of x from 0 excedes 1.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is.

The lesson uses a Desmos Activity entitled Guess My Rule. Students will give an input to a "machine" and the machine will give an output. Students will then be asked to determine a rule that the machine uses to produce an output from the given input. The goal of this activity is to develop the definition of a function using examples and nonexamples.