MCF3M Functions & Applications · Unit 3 — Quadratic Functions: Models and Forms

Lesson 3.1

Exploring a Situation Using a Quadratic Function

Use tables, scatter plots and second differences to recognize quadratic situations, build simple models, make predictions and choose sensible domains.

Estimated time: 30–45 minutes

Lesson progress0%

Learning goals

identify independent and dependent variables
create tables and scatter plots
calculate first and second differences to recognize quadratic relationships
use a quadratic model to make predictions
explain domain and range restrictions in real situations

Step 1 of 4

Warm Up

Success criteria

I can name the independent and dependent variables • find first and second differences • tell linear, quadratic and neither apart • predict with an equation • choose a domain that makes sense.

Warm Up

x: 0, 1, 2, 3 y: 2, 5, 10, 17

Classify the relationship in this table.

aLinearbQuadraticcNeither

Warm Up

Maya's pay depends on the hours she works. Which is the independent variable?

aHer paybThe hours she workscBothdNeither

Step 2 of 4

Learn

Real-world idea

Planning square gardens

A community group is building square garden plots. Bigger sides mean more growing space — but how fast does the area grow?

If the side doubles, does the area double too? Let's find out with a table.

Variables

The independent variable (x) is the one you choose: here, the side length in metres.

The dependent variable (y) depends on it: the area in square metres. We write A(x) = x².

Example A — Square garden areas, A(x) = x²
side x (m)	area A (m²)
1	1
2	4
3	9
4	16
5	25

Differences for A(x) = x²

Area: 1, 4, 9, 16, 25x goes up by 1 each time
First differences: 3, 5, 7, 9not constant → not linear
Second differences: 2, 2, 2constant → quadratic

Scatter plot of the garden areas

A(x) = x² for x = 1, 2, 3

When x-values are equally spaced, constant second differences mean the relationship is quadratic.

Constant FIRST differences mean linear.

Worked example

Use A(x) = x² to predict the area of a garden with 8 m sides, and give a sensible domain.

Substitute
Replace x with 8.
A(8) = 8² = 64
Interpret
The garden has an area of 64 m². Doubling the side from 4 m to 8 m made the area 4 times bigger (16 → 64).
Domain
A side length must be positive and can be any decimal, like 2.5 m. So the domain is x > 0 — continuous. The range is A > 0.

Answer: 64 m²; domain x > 0

Discrete or continuous?

Measured quantities (length, time, mass) are continuous — join the points with a smooth curve. Counted quantities (people, seats, sections) are discrete — plot separate points only.

Try it

A square garden has area A(x) = x², where x is the side length in metres. Predict the area when x = 8.

Your answer (m²)

Worked example

Theatre seating: S(n) = n² + 3n gives the seats in section n. Show the pattern is quadratic and describe the domain.

Table
n = 1, 2, 3, 4, 5
S = 4, 10, 18, 28, 40
First differences
Not constant, so not linear.
6, 8, 10, 12
Second differences
Constant, so quadratic.
2, 2, 2
Domain
n counts sections, so n = 1, 2, 3, … — positive whole numbers only (discrete).

Answer: Quadratic; domain: positive whole numbers

Try it

Theatre sections follow S(n) = n² + 3n seats, where n is the section number. How many seats are in section 6?

Your answer

Quadratic Pattern Detective

Investigate each dataset: plot it, finish its difference table, then decide what kind of relationship it is.

Dataset 1 — Bike rental

A bike rental costs a $2 helmet fee plus $3 per hour.

x-axis: hours · y-axis: cost ($)

x	0	1	2	3	4
y	2	5	8	11	14

First differences (next y − this y)

Second differences

Try it

For the square garden A(x) = x², which domain makes sense?

aAll real numbersbx > 0, any real lengthcx = 1, 2, 3, … onlydx < 0

Exploring a Situation Using a Quadratic Function

Warm Up

Learn

Planning square gardens

Variables

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 3.1 mastery check