Math Visualization — AI chat that plots functions for you

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Show and explain me the difference between linear and exponential growth. Give a concrete example and visualize both functions on the same plot so I can see where they diverge.

GPT-5.4

Here's the key idea:

A linear function changes by a constant amount each step. An exponential function changes by a constant factor each step.

Example

Function	Formula	How it grows
Linear	$L(x) = 2x + 1$	+2 for every step of $x$
Exponential	$E(x) = 2^{x}$	×2 for every step of $x$

At first, the linear function is ahead — but once $x$ is large enough, exponential growth always overtakes linear growth, no matter how steep the line is.

Linear vs Exponential Growth

When to use which

Linear — models a fixed rate (e.g. saving the same amount every month).
Exponential — models a rate proportional to the current value (e.g. compound interest, viral growth, bacterial populations).

Tip: if you double $x$ and the output doubles too, you have a linear relationship. If doubling $x$ squares the output, you're looking at exponential behaviour.

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