## Using a Calculator for Finding the Equation of a Function

To determine the function of best fit for a set of data, students should recognize which category of function bests fit the data and know how to use technology to obtain a function. This lesson teaches these skills and prepares students for the subsequent lesson(s), in which they will collect their own data.

### Part 1: Review of Graphs

Students should recognize which type of function a graph represents, so they can use technology to determine the best function for the data set. It is a worthwhile to review these functions since students may not have seen some of them in a while. Also, displaying all the graphs together, as this activity does, can aide students later, as they have to determine the best graph for the data.

Distribute copies of the Determine the Function review sheet, or use it as an overhead. Ask students to match each graph with its proper general description.

Determine the Function Overhead

### Part 2: Using the Technology

Whether students are using a calculator or a spreadsheet to do regression, this section is designed to introduce or review the process of finding regression functions.

Present students with the following table of data for the height of a crop of corn over time. This table also appears on Activity Sheet 1.

Activity Sheet 1: Height of Corn Over Time

### General Steps for Using Regression

(Specific TI-83 and TI-84 instructions are available here)

- Enter the data points into the calculator
- View the scatter plot of the data.
- Most likely, the window for viewing the graph is x={-10,10}and y={-10,10}. When the students first enter the data, and look at the scatter plot they will notice that only 4 of the 5 data points are visible. This is great time to discuss determining the appropriate domain and range for a specific set of data.

- Determine the best function category for the available data.
- Students may have different opinions about the best category from the points given. Discuss that in some situations, you may want to try more than one function to determine which is most appropriate for the data.

- Find the function of best fit for the data.
- Graph the function of best fit over the data to verify its reasonableness.

Allow some time for the students to answer the questions on Activity Sheet 1. Students may use the graph or the equation to help find their answers.

You may want to ask students some questions about the activity, such as:

- How do you think the calculator finds the equation of the line? Explain how you would get the equation of the line by hand?
- What makes the calculator easy to use?
- What do you have to remember in order to repeat the process?
- What did you write as a response to Question 11?

### Part 3: Finding the __Best__ Function

Sometimes it may not be clear from the given data points which regression to use. This example will elicit a discussion of “best” function.

Activity Sheet 2: Finding the Best Function

- Distribute Activity Sheet 2, Finding the Best Function.
- Students can work in pairs or small groups to complete the activity sheet.
- Different groups will get different answers. Most students will
determine that the function is exponential because they only see the
increase for the positive
*x*-values. - Encourage students to experiment with other functions until they
choose a quadratic form. They should notice that since the negative
*x*-values are unknown, it may be tricky to determine the*best*function.

- Different groups will get different answers. Most students will
determine that the function is exponential because they only see the
increase for the positive
- Discuss the results a whole class.
- What function category did you determine was best? Why?
- What would help determine which function is better? [Knowing more points or knowing the context or trend for the data.]

Finding the Best Function Answer Key

Once students can determine the function category and use technology to find the specific function, they are ready to explore the next lesson and determine the function resulting from various experiments.

- Graphing calculator
- Overhead - Determine the Function
- Activity Sheet - Corn Height
- Activity Sheet - Which Function Is Better?

**Assessment Options**

- Determine the graphs and functions for these data tables:

- Write out the steps to find the function from a data table, as though you were writing a tutorial. (
*A well-written description will be useful to students if they forget the process.*)

**Extensions**

- Finding the equation for a logarithmic function can be a bit more work,
since calculators don't often allow for different bases. Students will
have to determine the function for the inverse of the function
(switching
*x*and*y*) and then use the base to write the logarithmic function. - Move on to the next lesson,
*What's the Function?*

**Teacher Reflection**

- How eager were students to use a graphing calculator to find the functions that could represent the data?
- Describe the conversations you overheard between students as they discussed the procedures and strategies to find an equation to model a given set of data. How effective were students in helping each other find a mathematical model to represent the data?
- How well did students’ understanding of the material meet your expectations?
- What will you do differently the next time you present this lesson?

### Determining Functions Using Regression

### What's the Function?

### Learning Objectives

By the end of this lesson, students will:

- Analyze data to determine the type of function that most closely fits the data.
- Use a calculator to find the curve of best fit for a set of data.

### NCTM Standards and Expectations

- Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.

- Approximate and interpret rates of change from graphical and numerical data.