Direct Proportion: $y = k x$

Article Contents

What is Direct Proportion?
The Constant of Proportionality k
Table of Values
Graph of Direct Proportion
Positive vs Negative k
Comparing Different Values of k
Properties of Direct Proportion
Real-Life Examples
Common Mistakes
Interactive Exercises

1. What is Direct Proportion? {#what-is-direct-proportion}

Direct proportion is a function of the form:

f (x) = k x, k \neq = 0

It describes the relationship "when $x$ doubles, $y$ doubles too." The variable $y$ is directly proportional to $x$ .

Key feature

The graph of a direct proportion is a straight line through the origin $(0, 0)$ .

This makes sense: when $x = 0$ , $f (0) = k \cdot 0 = 0$ .

2. The Constant of Proportionality k {#constant-k}

The number $k$ is called the constant of proportionality (or slope).

k = \frac{y}{x} = \frac{f ( x )}{x}

Value of $k$	Effect
$k > 0$	$y$ increases as $x$ increases (graph goes up-right)
$k < 0$	$y$ decreases as $x$ increases (graph goes down-right)
$	k	$ large	Steep line
$	k	$ small	Gentle line

Finding k from a point

If you know that $f (3) = 12$ , then:

k = \frac{12}{3} = 4 ⟹ f (x) = 4 x

3. Table of Values {#table-of-values}

Example: $f (x) = 2 x$

$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$f (x)$	$- 6$	$- 4$	$- 2$	$0$	$2$	$4$	$6$

Notice: the ratio $\frac{f ( x )}{x} = 2$ is constant for every non-zero $x$ .

Example: $f (x) = - x$

$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$f (x)$	$3$	$2$	$1$	$0$	$- 1$	$- 2$	$- 3$

Here $k = - 1$ : the output is always the negative of the input.

4. Graph of Direct Proportion {#graph}

Every direct proportion $f (x) = k x$ is a straight line through the origin.

All three lines pass through the origin $(0, 0)$ .

5. Positive vs Negative k {#positive-vs-negative}

When $k > 0$ (e.g. $y = 2 x$ )

The function is increasing -- as $x$ grows, $y$ grows
The graph goes from bottom-left to top-right
Both variables change in the same direction

When $k < 0$ (e.g. $y = - x$ )

The function is decreasing -- as $x$ grows, $y$ shrinks
The graph goes from top-left to bottom-right
The variables change in opposite directions

6. Comparing Different Values of k {#comparing-k}

Function	$k$	Steepness	Direction
$y = 3 x$	$3$	Very steep	Increasing
$y = 2 x$	$2$	Steep	Increasing
$y = 0.5 x$	$0.5$	Gentle	Increasing
$y = - 0.5 x$	$- 0.5$	Gentle	Decreasing
$y = - 2 x$	$- 2$	Steep	Decreasing

Rule of thumb: The larger

∣ k ∣

, the steeper the line.

7. Properties of Direct Proportion {#properties}

Property	Value
Formula	$f (x) = k x$ , $k \neq = 0$
Graph	Straight line through origin
Domain	$D (f) = R$
Range	$R (f) = R$
y-intercept	Always $(0, 0)$
Increasing/decreasing	Increasing if $k > 0$ , decreasing if $k < 0$
Key ratio	$\frac{y}{x} = k$ is constant

8. Real-Life Examples {#real-life-examples}

Price and quantity

Pencils cost \$1.50 each. The total price:

P (n) = 1.5 n

Pencils $n$	$1$	$2$	$5$	$10$
Price $P$	\$1.50	\$3.00	\$7.50	\$15.00

Speed and distance

A runner moves at 8 km/h. Distance after $t$ hours:

d (t) = 8 t

Currency conversion

If 1 EUR = 1.10 USD:

USD (x) = 1.10 x

where $x$ is the amount in EUR.

Weight on the Moon

An object's weight on the Moon is about $\frac{1}{6}$ of its weight on Earth:

W_{Moon} (x) = \frac{1}{6} x \approx 0.167 x

9. Common Mistakes {#common-mistakes}

Mistake	Correction
Confusing direct proportion with linear function	Direct proportion is $y = k x$ (no constant term). Linear is $y = k x + q$
Thinking $k$ must be positive	$k$ can be negative (e.g. $y = - 3 x$ )
Forgetting the graph must pass through the origin	If the line does not pass through $(0, 0)$ , it is not a direct proportion
Confusing with inverse proportion	Direct: $y = k x$ (multiply). Inverse: $y = k / x$ (divide)

10. Interactive Exercises {#interactive-exercises}

Practice direct proportion:

Summary

Concept	Description
Direct proportion	$f (x) = k x$ , $k \neq = 0$
Graph	Straight line through the origin
$k > 0$	Increasing function
$k < 0$	Decreasing function
**$	k	$ large**	Steep line
Key property	$\frac{y}{x} = k$ is always the same

Introduction to Functions -- what is a function
Functions and Graphs -- plotting and reading graphs
Linear Function -- generalisation $y = k x + q$
Inverse Proportion -- the other type of proportion
Summary of Functions -- quick reference card

Direct Proportion

Direct Proportion: y=kx