LIVEx = 2.0 → y = 4.0 · Direct proportion: y = 2x

Mathematical Relationships

See equations become connected motion

Teacher mode

Mission: connect equation and graph

Move x and change the constant. Watch the point, table, equation, and caption update together.

Relationship family

Variables and parameters

Independent variable x2.0

Constant k2.0

Simulation speed1.0×

y = 2x

Live relationship

2.00input x

4.00output y

2.00instant rate

0.00y-intercept

Value table

x	y	change

Moving point

0x-coordinate

0y-coordinate

Camera

Zoom1.0×

What is happening?

In direct proportion, y changes by the same factor as x and the graph passes through the origin.

Teacher tools

Interactive visual lecture

Step 1 of 100%

Ready. Audio begins only when you press Start.

LanguageNarrator

Narration speed1.0×

Narration plays through embedded page audio at high volume, so browser tab recording can capture it. No microphone is used.

1. Variables are connected

What to watch

Watch the input point drive the output through a visible connection.

Visual cue sequence

Chapters

Learning objectives

Identify independent and dependent variables.
Interpret slope, intercept, constant of proportionality, and turning point.
Connect equations, tables, graphs, and real-time values.
Compare direct, linear, inverse, and quadratic relationships.

Applied formulas

Direct proportion: y = kx

Linear: y = mx + b

Inverse: y = k ÷ x

Quadratic: y = ax² + bx + c

Slope: change in y ÷ change in x

Suggested experiments

Reverse the sign of the slope and observe the graph direction.
Move the intercept without changing slope.
Approach x = 0 in the inverse relationship.
Change the sign of a quadratic coefficient.

Questions for exploration

Which relationships have a constant rate of change?
How can an equation predict a point not yet plotted?
What graph features reveal parameters immediately?

Real-world applications

Mathematical relationships model price and quantity, distance and time, scaling, pressure and volume, projectile paths, growth, optimization, and many other systems.

Common misconceptions

Every straight line is not a direct proportion; direct proportion must pass through the origin.
Intercept and slope describe different graph features.
Inverse does not mean negative.
A quadratic rate of change is not constant.

Relationships check

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