Exponential functions aˣ — graphs, properties, asymptote: Visual walkthrough

Every symbol below is defined before it is used. If the parent leaned on something, we rebuild it here.

Step 1 — Start with plain multiplication

WHAT. Pick a fixed number to multiply by. Call it the base $a$ . Start at the height $1$ and multiply by $a$ each time you take one step to the right.

WHY. The word exponential just means "repeated multiplication turned smooth." Before we can talk about a curve, we need the raw pattern of dots the curve will pass through. Multiplication (not addition) is the one choice that makes everything that follows happen.

PICTURE. Take $a = 2$ . Starting at $1$ and stepping right: $1, 2, 4, 8, 16$ . Each dot is twice as tall as the one before it.

Figure — Exponential functions aˣ — graphs, properties, asymptote

$start height f (0) = 1, next step f (x + 1) = fixed factor a \times current height f (x)$

$f (x)$ — the height of the curve at position $x$ .
$a$ — the fixed factor we multiply by at every step (here $2$ ).
The equation says: to move one step right, scale the height by $a$ .

Step 2 — Step LEFT instead of right: division

WHAT. Moving one step left must undo one step right. Stepping right multiplies by $a$ , so stepping left must divide by $a$ .

WHY. We want one single rule that works for the whole line, not just the right half. The only way "left then right returns you home" is if left-steps divide by exactly what right-steps multiply by. (This is the same index law $a^{m + n} = a^{m} a^{n}$ read backwards.)

PICTURE. From the start height $1$ , stepping left gives $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$ — each dot is half the previous. The dots shrink but stay above the floor.

$f (- 1) = \frac{f ( 0 )}{a} = \frac{1}{a}, f (- 2) = \frac{1}{a ^{2}}, f (- N) = a positive fraction \frac{1}{a ^{N}}$

$N$ — a whole number counting how many steps we have gone to the left.
$a^{N}$ — the base multiplied by itself $N$ times (a big number when $a > 1$ ).
$\frac{1}{a ^{N}}$ — one divided by that big number: a small positive number.

Step 3 — Fill in the gaps: the curve between the dots

WHAT. So far we only have dots at whole-number positions. A function $a^{x}$ works for every real $x$ — including $x = \frac{1}{2}$ , $x = 1.41$ , $x = 2$ . We connect the dots into a single smooth curve.

WHY. We do not draw a random wiggly line — we demand the multiplication law $a^{m + n} = a^{m} a^{n}$ hold for the in-between points too. That single demand forces $a^{1/2} = a$ (because $a^{1/2} \cdot a^{1/2} = a^{1} = a$ ) and pins down every gap by continuity. The curve is the only smooth line obeying "one step right = multiply by $a$ " everywhere.

PICTURE. The dots from Steps 1–2 sit on one gentle, ever-rising curve. It is smooth — no corners, no jumps.

$a^{1/2} \cdot a^{1/2} = a^{1/2 + 1/2} = a^{1} = a ⟹ a^{1/2} = a$

The half-step $x = \frac{1}{2}$ lands at height $a$ (for $a = 2$ , that's about $1.41$ ) — exactly between $1$ and $2$ , as the picture shows.

Step 4 — Two anchors every such curve shares

WHAT. No matter which base $a$ we chose, two points are guaranteed: $(0, 1)$ and $(1, a)$ .

WHY. $(0, 1)$ because taking zero steps leaves the start height untouched: $a^{0} = 1$ . $(1, a)$ because one step multiplies once: $a^{1} = a$ . These anchors let us read off the base straight from the graph — the height at $x = 1$ is the base.

PICTURE. Two curves, $y = 2^{x}$ (cyan) and $y = 5^{x}$ (amber), both pierce $(0, 1)$ , then split: at $x = 1$ one reaches $2$ , the other reaches $5$ .

$shared start (0, 1) a^{0} = 1 base revealed at (1, a) a^{1} = a$

Step 5 — The right-hand behaviour: unbounded growth

WHAT. Look right, where $x \to + \infty$ (that arrow means " $x$ marches off to the far right forever"). For $a > 1$ , the height grows without any ceiling.

WHY. Each rightward step multiplies by $a > 1$ , so heights form $1, a, a^{2}, a^{3}, \dots$ — a runaway staircase. There is no largest value; name any height and enough steps will pass it. This is what "explodes" means, precisely.

PICTURE. The cyan curve rockets up the right side of the frame; a dashed arrow shows there is no top.

$as x \to + \infty : a^{x} \to + \infty (for a > 1)$

$\to$ — "approaches / heads toward."
$+ \infty$ — not a number, but shorthand for "grows beyond every bound."

Step 6 — The left-hand behaviour: the floor appears

WHAT. Now look left, $x \to - \infty$ . Using Step 2, write $x = - N$ with $N \to + \infty$ . The height is $\frac{1}{a ^{N}}$ , which slides down toward $0$ .

WHY. This is the payoff. For $a > 1$ , the denominator $a^{N}$ explodes (Step 5's staircase). One divided by an exploding number collapses toward zero — but stays positive, because a positive over a positive is positive. So the curve dives toward the line $y = 0$ yet floats permanently above it.

PICTURE. Zoom in on the left tail: the curve grazes the $x$ -axis (dashed amber line $y = 0$ ), getting arbitrarily close, never landing.

$a^{- N} = \frac{1}{a ^{N}} N \to \infty 0^{+}$

$a^{- N}$ — the height $N$ steps to the left of the start.
$\frac{1}{a ^{N}}$ — a fraction with a huge positive bottom.
$0^{+}$ — approaching $0$ from above (the little plus means "stays positive").

Step 7 — Why the floor is never crossed

WHAT. Could the curve ever equal $0$ , or dip below? Solve $a^{x} = 0$ .

WHY. An asymptote is only meaningful if the curve truly never touches it. We must show no $x$ makes $a^{x}$ zero or negative. Every value $a^{x}$ is a positive number multiplied/divided by positive numbers — always positive. There is no exponent that produces $0$ .

PICTURE. The forbidden region ( $y \leq 0$ ) is shaded out; the curve lives entirely in the strip $y > 0$ .

$a^{x} = \frac{1}{a ^{N}} \neq = 0 (a nonzero-bottom fraction is never zero)$

So the range is strictly $y > 0$ : the floor $y = 0$ is a limit, not a value.

Step 8 — The degenerate mirror: decay ( $0 < a < 1$ )

WHAT. What if the base is a fraction, like $a = \frac{1}{3}$ ? Then stepping right shrinks the height. This is the mirror image of growth.

WHY. We must cover every case. Rewrite $(\frac{1}{3})^{x} = 3^{- x}$ : this is $3^{x}$ with $x$ replaced by $- x$ , i.e. the growth curve flipped in the $y$ -axis (see Graph transformations). Now the roles of "left" and "right" swap: the right tail hugs the floor, the left side explodes. The asymptote $y = 0$ survives — it is just approached from the other side.

PICTURE. $3^{x}$ (cyan, growth) and $(1/3)^{x} = 3^{- x}$ (amber, decay) reflected across the vertical $y$ -axis, both flattening toward the same floor $y = 0$ .

$(\frac{1}{a})^{x} = a^{- x} ⟹ reflection of a^{x} in the y -axis$

Case	Direction that grows	Direction that hugs $y = 0$
$a > 1$ (growth)	right ( $x \to + \infty$ )	left ( $x \to - \infty$ )
$0 < a < 1$ (decay)	left ( $x \to - \infty$ )	right ( $x \to + \infty$ )

The one-picture summary

This last figure compresses the whole walkthrough: the shared anchors, the exploding side, the hugging side, the forbidden region, and the asymptote — all at once.

Recall Feynman retelling — say it to a 12-year-old

Picture a number that doubles every step to the right: 1, then 2, 4, 8 — it takes off like a rocket, no top. Now walk backwards: half, a quarter, an eighth, a sixteenth. It keeps shrinking, but you always have some left — a crumb, then a speck, then a speck of a speck — never actually nothing. That "never quite zero" ground is the asymptote, the line $y = 0$ . Every one of these curves, whatever number you double or triple, walks through the same doorway at height $1$ when you take zero steps. And if instead of doubling you halve each step, it's the exact same picture held up to a mirror — the rocket and the crumb just swap sides, but the floor is still there, still never touched.

Connections

Exponential functions aˣ — graphs, properties, asymptote — the parent this walkthrough derives.
Index laws — the $a^{m + n} = a^{m} a^{n}$ demand that forced every step.
The number e and natural exponential eˣ — the special base where the growth rate equals the height.
Logarithms as the inverse of exponentials — reflect the finished curve in $y = x$ .
Graph transformations — the $y$ -axis reflection that turns growth into decay.
Exponential growth and decay models — where this floor-hugging behaviour shows up in the real world.