Intuiting Functions

by Justin Skycak (@justinskycak) on

A function is a scribble that crosses each vertical line only once.

This post is part of the series An Intuitive Primer on Calculus.

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A function is a scribble that crosses each vertical line only once.

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The instructions for drawing a function are given by the function’s equation. Each vertical line is labeled with a number, and when you plug that number into the $x$ variable in a function, the result tells you how high the function should be when it crosses that line.

For example, to draw the function whose equation is $y=3x+1$, we can start by plugging in the numbers $-1,$ $0,$ and $1.$

$\begin{align*} x=-1 \Longrightarrow y&=3(-1)+1=-2 \\[5pt] x=0 \Longrightarrow y&=3(0)+1=1 \\[5pt] x=1 \Longrightarrow y&=3(1)+1=4 \end{align*}$


Our results mean:

  • On the line $x=-1,$ the scribble needs to cross at height $y=-2.$
  • On the line $x=0$, the scribble needs to cross at height $y=1.$
  • On the line $x=1,$ the scribble needs to cross at height $y=4.$

In other words, we need to plot the points $(-1,-2),$ $(0,1),$ and $(1,4)$ on the grid. Since this is a linear function, we can draw the rest of the function just by connecting the dots.

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Linear functions are straight lines, but in general, functions tend to be curvy. There are many different types of functions, each of whose scribble curves in a different way.

This post is part of the series An Intuitive Primer on Calculus.


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