Proportional Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proportional Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A proportional function has the form f(x) = kx for a constant k \neq 0 — it passes through the origin and the ratio f(x)/x = k is constant.

Double the input, double the output. No offset—starts at zero.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proportional means \frac{y}{x} = k is constant. All points have same ratio.

Common stuck point: y = 2x + 5 is linear but NOT proportional (doesn't go through origin).

Sense of Study hint: Check: does the graph pass through (0, 0)? If not, it is linear but not proportional. Also check if y/x is the same for every data point.

Worked Examples

Example 1

easy
The weight of water is proportional to its volume. 5 liters weighs 5 kg. Write the function, find the constant of proportionality k, and compute the weight of 8.5 liters.

Solution

  1. 1
    Proportional function: W(V) = kV.
  2. 2
    Find k: W(5) = k \cdot 5 = 5 \Rightarrow k = 1 kg/L.
  3. 3
    Compute: W(8.5) = 1 \times 8.5 = 8.5 kg.

Answer

W(V) = V; k = 1 kg/L; W(8.5) = 8.5 kg
Direct proportionality y=kx means the ratio y/x is constant. Here the density of water is 1 kg/L, making it a clean example where k=1.

Example 2

medium
Determine whether y is proportional to x given the table: x: 2, 4, 6 and y: 7, 14, 21. If yes, find k and the equation.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Hooke's Law states spring force is proportional to stretch: F = kx. If F = 30 N when x = 6 cm, find k and the force when x = 10 cm.

Example 2

medium
Explain why f(x) = 3x + 2 is NOT a proportional function, and find the value of b that would make g(x) = 3x + b proportional.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

linear functionsproportionality