Visual walkthrough — Types — constant, linear, quadratic, polynomial, rational, radical, piecewise

Every function type in the parent note looks the way it does for one reason: the way its rate of change behaves. This page derives the whole family from absolute zero, one picture at a time. We start with a machine that never changes, and by the end we build up to curves that fly off to infinity.

Before line one, three words we will use constantly:

We only need the idea of "nudge $x$, watch the height." That's it. No calculus notation yet.

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Step 1 — The machine that ignores you: the constant

WHAT. Build the simplest possible rule: whatever number you put in, the same number $c$ comes out. Written $f(x)=c$.

$f(x)=\underset{\text{the only value ever returned}}{\underbrace{c}}\text{(no }x\text{ appears on the right!)}$

WHY. Notice there is no $x$ on the right side. That single fact means: nudging $x$ changes nothing. So the rate of change is $==0==$ everywhere.

PICTURE. Feed in $x=-2,0,3$ — all three land at the same height $c$. Connect them and you get a flat horizontal line (the blue line below). The vertical drop for any horizontal step is zero — that is flatness.

Constant rate $=0$ → flat line. Remember this: the rate of change is the thing that shapes the graph. Every later step just makes that rate more interesting.

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Step 2 — Let the rate be a fixed number: the line

WHAT. Instead of "rate $=0$", demand "rate $=m$, some constant number." Every time $x$ goes up by $1$, the height goes up by exactly $m$.

$\underset{\text{"height change}÷\text{ input change"}}{\underbrace{\frac{\Delta y}{\Delta x}}}=\underset{\text{a fixed number, the slope}}{\underbrace{m}}$

Here $\Delta$ (the Greek letter "delta") is shorthand for "the change in." So $\Delta x$ = "how far I stepped sideways," $\Delta y$ = "how far the height moved."

WHY. We choose a constant rate because that is the very next-simplest thing after zero. Rearranging, one sideways step of size $\Delta x$ always produces the same rise $\Delta y=m\Delta x$. Start from the height $b$ at $x=0$ and walk $x$ units right:

$y-\underset{\text{starting height at }x=0}{\underbrace{b}}=\underset{\text{rise per step}}{\underbrace{m}}\cdot \underset{\text{how far we walked}}{\underbrace{(x-0)}}\Rightarrow f(x)=mx+b$

PICTURE. Look at the yellow staircase: each tread is width $1$, each riser is height $m$. Because every step is identical, the corners line up perfectly straight — that is why constant rate forces a straight line. Change $m$ and the staircase tilts; change $b$ and the whole staircase slides up or down.

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Step 3 — Let the rate itself change: the quadratic bends

WHAT. Now make the rate not fixed but linear: the steepness grows like $2ax+b$. Accumulate that changing rate and you get:

$f(x)=\underset{\text{how strongly it bends}}{\underbrace{a}}{x}^2+\underset{\text{tilt at }x=0}{\underbrace{b}}x+\underset{\text{height at }x=0}{\underbrace{c}}$

WHY. In Step 2 a constant rate gave a straight line. If the rate increases as you move right, each step is a little steeper than the last, so the graph must curve upward. A rate that changes linearly is the simplest curving rate — and adding up (accumulating) a linear rate gives a squared term. That $x^2$ is the fingerprint of "acceleration."

PICTURE. Watch the green curve: near the left the steps are shallow, near the right they are steep. The steepness is itself growing — the curve bows. The lowest point (the vertex) is where the rate momentarily passes through zero before turning positive.

Where is that vertex? Complete the square (regroup so the $x$-terms hide inside one perfect square):

$a{x}^2+bx+c=a{\left(x+\frac{b}{2a}\right)}^2-\frac{b^2}{4a}+c$

A squared thing is smallest when it equals zero, i.e. when $x=-\frac{b}{2a}$. That is the vertex's $x$-coordinate — the balance point of the parabola.

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Step 4 — Stack more bends: the polynomial

WHAT. Keep going. Let the rate change in more and more elaborate ways by allowing higher powers:

$f(x)=\underset{\text{leading term}}{\underbrace{a_n{x}^n}}+a_{n-1}{x}^{n-1}+\cdots +\underset{\text{tilt}}{\underbrace{a_{1}x}}+\underset{\text{height at }0}{\underbrace{a_0}}$

The biggest power $n$ is the degree. Each extra degree buys you one more possible turn in the curve — up to $n-1$ bends.

WHY. A degree-$n$ polynomial can wiggle up to $n-1$ times because that many turns is exactly what its changing rate permits. But far from the origin, the largest power $a_n{x}^n$ dwarfs all the smaller ones — a big number raised to a big power crushes everything else.

PICTURE. The figure shows a cubic (degree 3, two bends) and a quartic (degree 4, three bends). Zoom out and both curves flatten into the shape of their leading term alone — the coloured dashed guides.

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Step 5 — Divide one curve by another: the rational blow-up

WHAT. Take two polynomials and put one over the other:

$f(x)=\frac{\stackrel{\text{top polynomial}}{\overbrace{p(x)}}}{\underset{\text{bottom polynomial}}{\underbrace{q(x)}}}$

WHY. Something brand-new happens that never happened before: the bottom $q(x)$ can hit zero. Dividing by zero is forbidden, so at those $x$-values the function has no output at all. As $x$ creeps toward such a value, the denominator becomes tiny, and a fixed top divided by a tinier-and-tinier bottom rockets toward $\pm \infty$. That runaway line is a vertical asymptote.

PICTURE. Study $f(x)=\frac{1}{x-1}$. As the red curve approaches $x=1$ from the right the height explodes up; from the left it plunges down. The curve hugs the dashed vertical line at $x=1$ but never touches it. Far to the sides, the height flattens toward $0$ — a horizontal asymptote, set by the ratio of the leading terms.

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Step 6 — The two degenerate edge cases

WHAT. Two special situations break the smooth pattern and deserve their own picture.

Case A — the radical's wall. $f(x)=\sqrt{x}$ exists only where $x\ge 0$, because a real square root of a negative number does not exist. So the graph simply stops at $x=0$ — a hard left wall (the domain is cut in half).

Case B — the piecewise jump. A piecewise rule uses different formulas on different regions, e.g.

$f(x)=\{\begin{array}{ll}-1& x<0\\ 1& x\ge 0\end{array}$

Here the graph can jump with no warning — it need not be joined at all (a break in continuity).

WHY. These show the boundaries of "smooth curve" thinking: a radical restricts where the machine works; a piecewise rule lets the machine switch personalities at a boundary. Neither is exotic — both are just honest about where their rule applies.

PICTURE. Left panel: the blue half-curve of $\sqrt{x}$ starting dead at the origin, nothing to its left. Right panel: the yellow step that leaps from $-1$ to $+1$ at $x=0$ (open circle = "not included," filled circle = "included").

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The one-picture summary

All six types side by side, arranged by how their rate of change behaves — the single thread running through the whole derivation:

Recall Feynman retelling — say it like you'd explain to a friend

Imagine a little dot walking left to right, and ask only one question: how is its height changing right now?

• If the height never changes, you get a flat line — that's a constant.
• If it changes by the same amount every step, you get a straight tilt — that's a line.
• If the amount of change is itself changing steadily, the walk curves into a parabola — a quadratic.
• Let the change wiggle in fancier ways and you can bend the path several times — that's a polynomial, and the biggest power tells you how far off it flies at the edges.
• Divide one such curve by another, and wherever the bottom becomes zero the height explodes to infinity — that's a rational function with its vertical walls.
• Finally, some machines only work on part of the number line (radicals, which need non-negative insides) or switch rules partway across and jump (piecewise).

Six shapes, one idea: the graph is just a drawing of how fast the height is moving. Master the rate, and you already know the shape.

Recall Quick self-test

Constant function's rate of change is ::: zero everywhere, so its graph is a flat horizontal line. What single sign decides if a parabola opens up or down? ::: the sign of $a$ (up if $a>0$, down if $a<0$). Why does $\frac{1}{x-1}$ blow up at $x=1$? ::: the denominator hits zero there, and dividing a fixed number by something shrinking to zero grows without bound. Why does $\sqrt{x}$ have no graph for $x<0$? ::: a real square root of a negative number does not exist, so those inputs are outside the domain.

See also: M02.01 Function Definition · M04.01 Derivatives (the rate of change made precise) · M02.04 Function Transformations · M02.06 Inverse Functions · M05.01 Integration (accumulating a rate, as in Step 3).