Proportionality and Functions

Problem — How to model proportionality situations using functions and interpret their graphical representations?

Objectives

Understand the link between proportionality and functions.
Recognize and use a proportional function.
Distinguish a proportional function from an affine function.
Know how to read, complete, and interpret a value table.
Draw and utilize graphical representations of these functions.
Solve real-world problems using these models.

Key Points

A proportional function is written as f(x)=kx and its line passes through (0;0).
An affine function is written as f(x)=ax+b and its line passes through (0;b).
The proportional function is a special case of the affine function when b=0.
The coefficient k or a corresponds to the slope of the line.
The number b corresponds to the y-intercept, meaning the value at x=0.

Introduction

In mathematics, a function allows associating a value x with another value noted f(x). In 5th grade, we especially study two very important types of functions: the proportional function and the affine function.

These functions help model many everyday situations: prices, distance, speed, subscriptions, consumption, temperature, etc.

Important Definition

Modeling a situation means translating it into a mathematical expression to better understand, represent, and calculate it.

Introduction Summary

Functions are tools that allow connecting two quantities.
In 5th grade, one should know how to recognize whether a situation is proportional or affine.

Part 1: Proportional Function

Important Definition

A proportional function is a function defined by f(x) = k × x, where k is the proportionality coefficient.

Properties

Its graphical representation is a line passing through the origin (0;0).
The number k represents the change of the function as x varies.
For a proportional function, if x is multiplied by a number, then f(x) is multiplied by the same number.
The coefficient k is also the slope of the line.
We also have k = f(1).

Concrete Interpretation

In a proportionality situation, there is no fixed starting value. Everything depends directly on the chosen quantity.

For example, if 1 kg of fruit costs 3 €, then 2 kg cost 6 €, 4 kg cost 12 €: the price depends directly on the amount purchased.

Testing Proportionality

Coefficient Method: if y = k × x with the same k for all values, the situation is proportional.
Quotient Method: if the ratio y ÷ x is always the same (for x ≠ 0), the situation is proportional.
Cross-Product Method: for two pairs (x₁, y₁) and (x₂, y₂), check if x₁ × y₂ = x₂ × y₁.
Graphical Method: the graph of a proportional situation is a line that passes through (0;0).

Examples

f(x) = 4x: k = 4. If x doubles, then f(x) doubles as well.

Proportional price: 2 kg → 7 €, 5 kg → 17.50 €: 7 ÷ 2 = 3.5 and 17.50 ÷ 5 = 3.5 ⇒ proportional.

Verification by cross product: 2 × 17.50 = 35 and 5 × 7 = 35 ⇒ proportional.

x	f(x) = 2x	f(x) = 4x
0	0	0
1	2	4
2	4	8
3	6	12
5	10	20

Method

To quickly recognize a proportional function, follow these steps:

Check if the expression is of the form kx.
Check if f(0)=0.
Check if the line passes through the origin (0;0).
If needed, calculate the ratio y ÷ x to see if it stays constant.

Summary of Part 1

A proportional function is noted as f(x)=kx.
Its representation is a line that passes through (0;0).
The coefficient k is both the proportionality coefficient and the slope.

Part 2: Affine Function

Important Definition

An affine function is a function defined by f(x) = a x + b, where a is the slope and b the y-intercept.

Properties

Its graphical representation is a line.
This line does not necessarily pass through the origin.
It crosses the y-axis at point (0;b).
The value a indicates how f(x) changes when x increases by 1.
The value b is the function's value when x = 0.

Concrete Interpretation

An affine function often models a situation with:

a fixed part: this is b;
a variable part: this is a x.

Examples

f(x) = 3x + 2: a = 3 and b = 2. The line crosses the y-axis at 2.

g(x) = 0.05x + 10 models a cost: a fixed part of 10 € and a variable part of 0.05 € per unit of x.

For example, for x = 100, we get g(100)=0.05×100+10=15.

Link Between Proportional and Affine Functions

Every proportional function is also an affine function, but a special one.

Important Definition

A proportional function is a special case of an affine function where b = 0.

Type	Expression	Graph	Parameters
Proportional	`f(x) = kx`	Line through (0;0)	`k`: slope and coefficient
Affine	`f(x) = ax + b`	Line, crosses at `(0;b)`	`a`: slope; `b`: y-intercept

Method

To recognize an affine function, follow these steps:

Check if the expression is of the form ax+b.
Identify the value of b, which corresponds to f(0).
Observe if the line crosses the y-axis at (0;b).
Check if the situation has a fixed part and a variable part.

Summary of Part 2

An affine function is written as f(x)=ax+b.
It often models a situation with a fixed part + variable part.
If b=0, then the affine function becomes a proportional function.

Part 3: Distinguishing Proportional and Affine Functions

Quick Method

Look at the expression:
- kx ⇒ proportional
- ax+b ⇒ affine
Look at the value at x=0:
- if f(0)=0, it may be proportional;
- if f(0)=b with b ≠ 0, then it is not proportional.
Look at the graph:
- line passing through the origin ⇒ proportional;
- line not passing through the origin ⇒ non-proportional affine.

Examples

f(x)=5x is proportional because there is no added term and f(0)=0.

g(x)=5x+4 is affine but not proportional because g(0)=4.

Pitfalls

Not every line necessarily represents a proportional situation.
An affine function can have a line that goes up or down, but if it does not pass through (0;0), it is not proportional.
Do not confuse a and b:
- a corresponds to change;
- b corresponds to the starting value.
A function of the form ax is both affine and proportional: it is not one or the other, but both.
A line parallel to that of a proportional function is not necessarily proportional.

Summary of Part 3

The simplest criterion is: passes through origin ⇒ proportional.
An affine function has a y-intercept equal to b.

Part 4: Graphical Representation

Proportional Function

Its graph is a line passing through (0;0).
The coefficient k gives the slope.
To draw this line, two points are enough, for example (0;0) and (1;k).

Affine Function

Its representation is also a line.
It crosses the y-axis at (0;b).
First plot (0;b), then use the slope a.
If a=2, when x increases by 1, then y increases by 2.
If a=-1, when x increases by 1, then y decreases by 1.

Example

Plot f(x) = 2x and g(x) = 2x + 3: these two functions have the same slope 2. Their lines are therefore parallel. But g is shifted up by 3 because g(0)=3.

x	f(x) = 2x	g(x) = 2x + 3
0	0	3
1	2	5
2	4	7
3	6	9
4	8	11

Reading a Graph

Reading the y-intercept means reading the value of the function at x=0.
Reading the slope consists of observing how much y changes when x increases by 1.
A line that rises from left to right has a positive slope.
A line that falls from left to right has a negative slope.

Method

To read a function graph:

Check if the line passes through the origin or not.
Read the point where the line crosses the y-axis.
Observe how y changes as x increases by 1.
Deduce whether it is a proportional or affine function.

Summary of Part 4

Passes through the origin ⇒ proportional function.
Passes through (0;b) with b ≠ 0 ⇒ non-proportional affine function.
The slope describes how the function changes.

Part 5: Problem Solving

General Method

Identify the nature of the relationship: proportional or affine.
Determine the quantities involved and what they represent.
Write the appropriate expression: kx or ax + b.
Calculate the required values.
Interpret the result in the context of the problem.

Examples

Proportional — Constant speed: d(x)=60x where x is in hours and d in km. In 2.5 h: d(2.5)=150 km.

Affine — Subscription: f(x)=0.05x+10 where x is in minutes and f in euros. For 100 min: f(100)=15 €.

Detailed Example

A taxi charges 4 € for the initial fee then 2 € per kilometer.

If x is the number of kilometers driven, the price is given by P(x)=2x+4.

This situation is not proportional, because even when x=0, you already pay 4 €.

Pitfalls

A situation with a starting price, subscription, or fixed fees is generally not proportional.
Don't forget to interpret the meaning of x and f(x) in the problem.
A correct mathematical expression is not enough: you also have to verify that it corresponds well to the real situation.

Summary of Part 5

Functions are modeling tools.
The proportional function models a single variation.
The affine function models a fixed part + a variable part.

Final Course Summary

The proportional function (f(x)=kx) is a special case of the affine function (f(x)=ax+b) when b=0. In 5th grade, one must know how to recognize, distinguish, complete value tables, represent graphically, and use these functions to model real-life situations.

Proportionality and Functions

Introduction

Part 1: Proportional Function

Properties

Concrete Interpretation

Testing Proportionality

Part 2: Affine Function

Properties

Concrete Interpretation

Link Between Proportional and Affine Functions

Part 3: Distinguishing Proportional and Affine Functions

Quick Method

Part 4: Graphical Representation

Proportional Function

Affine Function

Reading a Graph

Part 5: Problem Solving

General Method

Aller plus loin : Quiz et exercices

Proportionnalité : reconnaître et utiliser k

Pourcentages : calculer, augmenter et réduire

Problèmes : proportionnalité ou pourcentage ?