 Downloadable versions of this lesson are available in the following formats:

## Overview

In this lesson, students will learn about applications of the vertex form [y = a(x – h)2 + k] of a quadratic equation. Students will extract data on births in Canada from Statistics Canada's E-STAT database and import them into a statistical software program. Within the software program, students will model a quadratic equation. By adjusting the values of the a, h, and k parameters to maximize the fit of the parabola to the Canadian Baby Boom data, students will gain a greater understanding of the purpose of the parameters in the vertex form of a quadratic equation.

Contributors: Manny Avila, Queen's University; Joel Yan and Jennifer Hall, Statistics Canada

## Objectives

• Through investigation, determine the relationships between the graphs and the equations of quadratic functions
• Through investigation, determine the basic properties of quadratic functions
• Identify the effect of simple transformations
• Explain the role of a, h, and k in the vertex form [y = a(x – h)2 + k] of the quadratic equation representing a parabola
• Collect secondary data that may be represented by a quadratic function
• Fit the equation of a quadratic function to a scatter plot using an informal process

Mathematics

## Duration

One to two 75 minute periods

## Materials

Computers with Internet access and statistical software
Computer projector
E-STAT account
E-STAT instructions
Software instructions:

• Generic
• Dynamic statistical Fathom software (PDF, 45 Kb)

## Prior knowledge

Vertex form of quadratic equation [y = a(x – h)2 + k]
Basic knowledge of E-STAT and statistical software

## Classroom instructions

1. Discuss important properties of the vertex form of the quadratic equation as a review.
2. Using the computer projector, demonstrate the important features of E-STAT.
3. Hold a brief class discussion on the topic of the Baby Boom to assess students' prior knowledge and share information on the topic.
4. Distribute the student instructions and worksheet and have students complete the lesson independently or in pairs.

## Enrichment

Have students repeat this process for a later time period to see if there is a significant Baby Boom Echo. Ask them to compare the shapes of the two curves, find the peak year of the Baby Boom Echo, find the period between the two peaks, and interpret this period of time.

Have students repeat this process for their province or territory instead of Canada as a whole. Ask them to compare the shapes of the graphs. If the shape of the graph for the province or territory is different from the shape of the graph of Canada as a whole, ask students to research reasons for this difference.

Have students import the data for the Baby Boom and the Baby Boom Echo into a graphing calculator or spreadsheet software to perform quadratic regression analysis. Have the students compare their curve of best fit for the data with the regression analysis.

Challenge your students to search on the E-STAT CANSIM database to find other time series data (among millions of time series) that can be modelled by a quadratic function. They can import these data into a statistical software program and attempt to plot quadratic functions to fit the data.

If students find an E-STAT time series that can be modelled well by a quadratic function, please e-mail us.

## Evaluation

Students can be informally assessed on their work habits and computer skills throughout this activity. They can be formally assessed via the worksheet, which can be marked using a marking scheme of the teacher's choice. 