Foundations — Exponential functions aˣ — graphs, properties, asymptote

This page assumes you know nothing. We build each symbol the parent note uses, one at a time, so that when you re-read it, every line is already yours. Read top to bottom — each block leans on the one before it.

1. A number line, and what "positive" and "negative" mean

Before any power, we need the stage everything lives on: the number line. It's a straight ruler stretching forever in both directions. The middle is $0$. To the right the numbers grow ($1,2,3,\dots$); to the left they go negative ($-1,-2,-3,\dots$).

Read the figure: the red dot marks $0$, the centre. Everything to its right is positive, everything to its left is negative. Notice the ruler has no ends — the two black arrows say it continues forever both ways.

A number is called real if it lives somewhere on this line — whole numbers like $3$, fractions like $\frac{1}{2}$, and the in-between numbers that never end like $1.41421\dots$ all count. When we later say the exponent $x$ can be "any real number", we mean $x$ can sit at any point at all on this line.

Why the topic needs this: the parent note constantly says "$a>0$", "$y>0$", "range is strictly positive". Those are just statements about which side of the line a number lives on. The symbol $>$ means "sits to the right of / is bigger than".

2. The two things $a$ and $x$ are allowed to be (state this up front)

Before we ever write $a^x$, pin down what each letter is permitted to be — the parent note assumes this, so we make it explicit here.

Why the topic needs this: every derivation below quietly relies on $a$ being positive and non-zero. Stating it now means no step later "cheats".

3. Multiplication as stretching, and the symbol $\cdot$

The heart of this whole chapter is multiplication. We write it two ways: $3\times 2$ or, in maths, $3\cdot 2$ (a raised dot). They mean the same thing.

Why the topic needs this: the parent's big idea is "multiply by the same factor each step." A "factor" is just the number you multiply by. Growth ($a>1$) = stretch outward each step; decay ($0<a<1$) = squash inward each step.

4. From repeated multiplication to a power: the symbol $a^x$

Now the star notation. Writing $a\cdot a\cdot a$ is clumsy. So we shorthand it (remember $a>0$, $a\ne 1$ from Section 2):

$a^3=a\cdot a\cdot a(\text{three copies of }a\text{ multiplied}).$

Read the figure: on the left, three identical black boxes each hold an $a$ — that's the repeated multiplication $a\cdot a\cdot a$. On the right is its shorthand $a^3$: the big black $a$ is the base (the brick), and the small red $3$ up top is the exponent telling you "use three copies". The red highlight is exactly the moving part in an exponential.

Why "up top" matters: the parent's mnemonic is "POWER ON TOP → SHOOTS OFF THE TOP." The variable that changes is the exponent (the thing on top). Compare:

5. The variable $x$ and functions $f(x)$

A variable is a letter that stands for "any number you like to feed in" — usually $x$, and (from Section 2) here it may be any real number. A function is a machine: put a number $x$ in, get exactly one number out. We name the machine $f$ and write the output as $f(x)$, read "$f$ of $x$".

Why the topic needs this: it lets us talk about the whole curve at once instead of one number. Every point on the graph is a pair (input, output) $=(x,f(x))$.

6. The radical symbols — square roots, cube roots, and $n$th roots

We're about to need one more family of symbols before the index law. When we ask "what number, multiplied by itself, gives $9$?" the answer is $3$, because $3\cdot 3=9$. That answer has a name and a symbol — and it has bigger cousins.

Sometimes we want a number multiplied by itself three times, or $n$ times, to give $b$. Same idea, with a small number tucked into the radical:

Why negative bases were banned (promised in Section 2): $\sqrt{-4}$ would need a number that squares to $-4$. But any real number squared is positive (a negative times a negative is positive), so no real answer exists. That's the "holes everywhere" the parent warned about — and exactly why we insisted $a>0$. Keeping $b>0$ means every root $\sqrt[n]{b}$ is a clean single positive number.

Why the topic needs this: in the very next section, the fraction exponent $a^{1/q}$ turns out to be the $q$th root $\sqrt[q]{a}$. Without $\sqrt[n]{}$ defined, general rational exponents like $a^{p/q}$ would be gibberish.

7. The index law — the single rule that defines everything

Here is the law the parent note says "we demand stays true" (throughout, $a>0$ so $a^n$ is a positive, non-zero number):

Picture why it's obviously true for whole numbers: $a^2\cdot a^3=(a\cdot a)\cdot (a\cdot a\cdot a)=a^5=a^{2+3}$. You just line the bricks up in a row and count.

One tool we'll reuse — dividing both sides of an equation. Several steps below "divide both sides by $a^n$". Here is why that's legal, stated from zero: an equation $L=R$ says two numbers are the very same number. If you do the identical operation to both — here, multiply both by $\frac{1}{a^n}$ — you still have the same number on each side, so equality survives. This is only safe when $\frac{1}{a^n}$ actually exists, i.e. when $a^n\ne 0$ — which is guaranteed because $a>0$ makes $a^n$ a positive, non-zero number. (You may never divide by $0$.) Keep that licence in mind for every "divide both sides" below.

Why the topic needs this — deeply: the parent uses this ONE law to define $a^x$ for exponents that aren't whole numbers. Watch how each awkward case is forced:

• $a^0=1$: demand $a^0\cdot a^n=a^{0+n}=a^n$. Now divide both sides by $a^n$ (legal, since $a^n\ne 0$). That leaves $a^0=1$. We didn't choose it — the law chose it. This is why every curve passes through $(0,1)$.
• $a^{-n}=\frac{1}{a^n}$: demand $a^{-n}\cdot a^n=a^0=1$, so $a^{-n}$ must be the number that multiplies $a^n$ to give $1$ — that's its reciprocal.
• $a^{1/2}=\sqrt{a}$: demand ${\left(a^{1/2}\right)}^2=a^{1/2+1/2}=a$. The thing that squares to $a$ is exactly the square root $\sqrt{a}$ from Section 6.
• $a^{1/q}=\sqrt[q]{a}$ (any whole $q\ge 1$): demand ${\left(a^{1/q}\right)}^q=a^{\frac{1}{q}+\frac{1}{q}+\cdots +\frac{1}{q}}=a^1=a$ (that's $q$ copies of $\frac{1}{q}$ adding to $1$). So $a^{1/q}$ must be the number that, raised to the power $q$, gives $a$ — which is precisely the $q$th root $\sqrt[q]{a}$ from Section 6.
• General rational $a^{p/q}$: now stack the two ideas. Since $\frac{p}{q}=p\cdot \frac{1}{q}$, applying the law repeatedly gives $a^{p/q}={\left(a^{1/q}\right)}^p={\left(\sqrt[q]{a}\right)}^p=\sqrt[q]{a^p}$. So every fraction exponent is a root of a power — fully defined, and real because $a>0$.

You'll meet these in full in Index laws. For now, just hold: one law, and $a^0=1$, negatives, roots, and all fractions fall out.

Filling the last gaps (irrational exponents): whole-number, negative and fraction exponents cover every rational point on the number line, but they still miss the endless-decimal ones like $\sqrt{2}=1.41421\dots$. For those, $a^{\sqrt{2}}$ is defined by squeezing — see the next section for the picture.

Read the figure: the exponent axis is drawn at the bottom, with $\sqrt{2}\approx 1.4142$ marked in red. We evaluate $2^x$ at rational exponents that trap $\sqrt{2}$ from both sides: from below $2^{1.4},2^{1.41},2^{1.414},\dots$ (black dots climbing up) and from above $2^{1.5},2^{1.42},2^{1.415},\dots$ (black dots coming down). The two chains of dots close in on a single height — the gap between the highest "below" value and the lowest "above" value shrinks toward $0$. Because the values are squeezed into a shrinking interval with only one number left inside, that number is forced and unique: we name it $2^{\sqrt{2}}$. This is why $a^x$ exists for every real $x$, not just fractions.

8. Reciprocal $\frac{1}{a}$ and the reflection idea

A reciprocal of $a$ is $\frac{1}{a}$ — the number that multiplies $a$ back to $1$. The reciprocal of $3$ is $\frac{1}{3}$; of $\frac{1}{3}$ is $3$.

You'll see the full mirror-and-shift toolkit in Graph transformations.

9. Coordinates and the graph — the symbol $(x,y)$

To draw a function we use two number lines crossed at right angles: the axes. The horizontal one is the $x$-axis (inputs), the vertical one the $y$-axis (outputs). A point is written $(x,y)$: go right $x$, go up $y$.

Read the figure: the black curve is $y=2^x$. The two red dots are the anchor points the parent note cares about: $(0,1)$ sits one unit up on the $y$-axis (because $2^0=1$), and $(1,2)$ sits at height $2$ above $x=1$ (because $2^1=2$ — the base itself). Trace left and the curve sinks toward the horizontal axis; trace right and it rockets up.

Why the topic needs this: the parent note's special points are all coordinates: $(0,1)$ means "input $0$, output $1$"; $(1,a)$ means "input $1$, output $a$". You cannot read those until $(x,y)$ is second nature.

10. Approaching but never reaching — $\to$, $\infty$, and "asymptote"

Two final symbols the parent leans on hard.

Read the figure: two curves share one red asymptote. The black growth curve $y=2^x$ (base $>1$) dives toward the red floor as you go left ($x\to -\infty$). The black decay curve $y={\left(\frac{1}{2}\right)}^x$ (base between $0$ and $1$) is its mirror image and dives toward the same red floor as you go right ($x\to +\infty$). The red horizontal line $y=0$ is the asymptote both hug but neither touches.

Case $a>1$ (growth). As $x\to -\infty$ write $x=-N$ with $N\to +\infty$: then $a^{-N}=\frac{1}{a^N}$, and since $a>1$ the denominator $a^N$ grows huge, so $\frac{1}{a^N}\to 0^{+}$. The curve hugs the floor $y=0$ on the left forever without landing — because a fraction $\frac{1}{\text{big positive}}$ is tiny but never exactly zero.

Case $0<a<1$ (decay). Now the picture flips. Write $a=\frac{1}{b}$ with $b>1$, so $a^x=b^{-x}$. As $x\to +\infty$ we get $b^{-x}=\frac{1}{b^x}$, and $b^x$ grows huge, so $a^x\to 0^{+}$. So a decay base still hugs the same floor $y=0$ — but now on the right ($x\to +\infty$) instead of the left. It's exactly the growth picture reflected in the $y$-axis (Section 8). Either way, the range stays $y>0$ and the asymptote is $y=0$, never crossed.

Why the topic needs this: the parent's most-tested fact — "range is $y>0$, asymptote is $y=0$, never crossed" — is entirely this idea, and it must hold for both growth and decay bases you were allowed back in Section 2.

Prerequisite map — and how to read it

The diagram below is a flow map: each box is one idea from this page, and an arrow "$A\to B$" means "you need $A$ before $B$ makes sense." Start at the top boxes (the things needing nothing) and follow arrows downward; everything funnels into the single bottom box, the exponential function itself. Use it as a checklist: if any box feels shaky, re-read its section above.

Once these feed together you're ready for the parent: Exponential functions aˣ — graphs, properties, asymptote. From there the natural next steps are The number e and natural exponential eˣ and Logarithms as the inverse of exponentials, and the applications in Exponential growth and decay models.

Equipment checklist

Test yourself — you should be able to say each answer before revealing it.