Introduction to Functions

Teacher’s Guide

The following lessons can be used as an introduction to functions in a Pre-Algebra or Algebra 1 class:

- What Is a Function?
- What Are the Domain and Range?
- What Is Function Notation?
- What Does It Mean to Evaluate a Function?
- Sequences and Functions
- What Is a Recursive Function?

These lessons can be shared with students by simply cutting and pasting the URLs and sharing with the class. Students don’t need a special log-in to access these lessons.

This Teacher’s Guide provides additional information on each of the four lessons, along with other Media4Math digital resources to use to supplement the content of these lessons.

As a group, these lessons will give your students the basics of understanding functions. These lessons include videos, clear graphics, definitions, and several use the Desmos graphing calculator tool to reinforce key concepts.

As a group, these lessons address the following Common Core standards:

This lesson introduces the concept of a function and focuses on the unique output value for each input value. Students are shown that two different inputs can have the same output, but a single input can’t have two outputs. This is reinforced through a function table and graphs of coordinates.

The main learning objectives include:

- Define a function
- Graph coordinate pairs
- Define the vertical line test

Finally, students are shown a Desmos graphing calculator window with a set of graphed coordinates. This set of coordinates do not form a function because they fail the vertical line test in two places. Students are asked to change the coordinates in the table to define a function.

Here is a link to that Desmos page: https://www.desmos.com/calculator/6yqgqv9vhy.

This lesson introduces the concept of defining the domain and range of a given function. In the previous lesson we looked at functions defined as a set of discrete coordinates. In this lesson we also look at the domain and range of a continuous function. We start with an example of a function defined by a line segment. This allows for the definition of the domain and range that are continuous but also not infinite. We then go on to the case of a linear function that extends to infinity.

The main learning objectives include the following:

- Find the domain and range for a set of coordinates that define a function
- Find the domain and range of a continuous function that doesn’t extend to infinity
- Find the domain and range of a continuous function that extends to infinity

Finally, students are asked to define the domain and range of a three functions in a Desmos graphing calculator window. Here is a link to the Desmos window: https://www.desmos.com/calculator/dip0vlfke0.

Because this is an interactive graph, you can have students redefined the domain to see the impact on the graph.

This lesson introduces function notation. In the previous lessons, functions were defined in terms of tables and graphs, as well as the vertical line test and function machine graphics. In this lesson we function notation to define a function algebraically.

With function notation comes the concept of evaluating a function and what that looks like in function notation. We show how to evaluate a linear function. By being able to define different values for x and f(x) we generate a function table and use it to graph coordinates. By showing coordinates in a function table, we can see the pattern in the values for f(x). This brings up an example of given a data table using the pattern to complete the table.

The main learning objectives include the following:

- Define a function using function notation
- Evaluate a function
- Graph a function based on function notation
- Vertically and horizontally displace a function

Finally, students are shown a Desmos graphing calculator window with a partially completed function table. They are asked to complete the table to graph the coordinates along the given linear pattern. Here is the link to the Desmos window: https://www.desmos.com/calculator/56zmsil0l9.

While evaluating a function was briefly introduced in the previous lesson, in this lesson we go into detail on evaluating a linear function and a quadratic function. Students are introduced to the general form of the slope-intercept form and the quadratic function in standard form. Students evaluate each type of function for different values of x.

The main learning objectives include the following:

- Define a function using function notation
- Evaluate a function using function notation
- Evaluate linear functions
- Evaluate quadratic functions

Finally, students are shown two Desmos graphing calculator windows one with a linear function in slope-intercept form and one with a quadratic in standard form. For each function sliders allow the students to vary values of the coefficients and constants. A function table is linked to each function and functions and coordinates are graphed.

Here are the links to the Desmos windows:

This lesson shows the connection between arithmetic sequences and functions. In this lesson students are shown an arithmetic sequence are are asked to find a function that can be used to generate the sequence.

The main learning objectives include the following:

- Analyze Arithmetic Sequences
- Derive a function in function notation from an arithmetic sequence

Finally, students are shown a Desmos graphing calculator window and are asked to define a function that overlaps the graph of the coordinates shown.

Here is the link to the desmos window: https://www.desmos.com/calculator/2xsmknesml.

This lesson shows the connection between arithmetic sequences, recursive functions, and explicit functions. Students are shown multiple representations of functions and are asked to generate an explicit function, given a recursive function and a corresponding arithmetic sequence.

The main learning objectives include the following:

- Define a recursive function
- Generate a sequence from a recursive function
- Define an explicit function
- Generate an explicit function from a sequence

Finally, students are shown two Desmos graphing calculator windows. In one students are asked to generate terms in an arithmetic sequence, for a given recursive function. In the second, given a recursive function, arithmetic sequence, and graph, students are asked to find the corresponding explicit function.

Here are the links to the Desmos windows:

Review the topic of functions with this video segment.

To test your students as you work your way through these lesson plans, here are some suggested assessments:

- Evaluating Linear Functions: Set of Digital Flash Cards.
- Evaluating Linear Functions: Interactive Game.
- Evaluating Linear Functions: Worksheet.
- Evaluating Quadratic Functions: Set of Digital Flash Cards.
- Evaluating Quadratic Functions: Interactive Game.
- Evaluating Quadratic Functions: Worksheet.

For students who need additional help graphing coordinates, have them try this worksheet.