Proportional Relationships and Graphs

Proportional relationships describe how one quantity changes in relation to another quantity. They help connect algebraic formulas, graph shapes, and patterns of change. The constant k represents the constant of proportionality.

The graphs below show several important ways that y can depend on x. Some relationships increase as x increases, while others decrease as x increases. Recognising these graph shapes helps make equations, tables, and visual patterns easier to understand.

Table of Proportional Relationships

Proportionality in words Using ∝ Formula Desmos URL
y is directly proportional to the square of x y ∝ x² y = kx² https://www.desmos.com/calculator/vhjtgkczw3
y is directly proportional to the cube of x y ∝ x³ y = kx³ https://www.desmos.com/calculator/2kovuylbbj
y is directly proportional to the square root of x y ∝ √x y = k√x https://www.desmos.com/calculator/v45uehjp4o
y is inversely proportional to x y ∝ 1/x y = k/x https://www.desmos.com/calculator/k6qvoh6r5b
y is inversely proportional to the square of x y ∝ 1/x² y = k/x² https://www.desmos.com/calculator/d4dcxrhlu7

Why These Graphs Matter

y = kx²

The graph of y = kx² is a parabola. It shows how y changes with the square of x. Squaring makes the output grow faster than a simple linear relationship.

This graph is useful for understanding curved growth, symmetry about the y-axis, and the effect of changing k.

y = kx³

The graph of y = kx³ is a cubic graph. It shows how y changes with the cube of x. Cubic graphs can increase or decrease very quickly as x moves away from zero.

This graph is useful for understanding odd powers, rotational symmetry about the origin, and the difference between square and cube relationships.

y = k√x

The graph of y = k√x is a square root graph. It shows growth that becomes slower as x increases.

This graph is useful for understanding restricted domains, since the square root graph usually starts at x = 0 when working with real numbers.

y = k/x

The graph of y = k/x is a reciprocal graph. It shows inverse proportionality: as x increases, y decreases.

This graph is useful for understanding asymptotes, reciprocal relationships, and why the graph does not pass through the origin.

y = k/x²

The graph of y = k/x² is a reciprocal square graph. It shows inverse proportionality to the square of x. The value of y changes more sharply near x = 0 than in the graph of y = k/x.

This graph is useful for comparing inverse relationships and understanding how powers affect the shape and behaviour of a graph.

Summary

These graph types show how changing the power or position of x changes the shape of a graph. They also help distinguish between direct proportionality and inverse proportionality.

Understanding these relationships builds stronger links between formulas, graph shapes, proportional reasoning, and mathematical modelling.

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