Exponential functions aˣ — graphs, properties, asymptote: Worked examples

Before anything: three things we will lean on.

$a^{x}$ means "the base $a$ multiplied together, but with the exponent $x$ allowed to be any real number" (see the parent for how fractions and negatives are defined).
$x \to - \infty$ is read " $x$ marches off to the far left forever". The arrow $\to$ means "gets closer and closer to".

The scenario matrix

Every exam question on $a^{x}$ is really one of these cells. Each worked example below is stamped with its cell.

Formula Case classes for

$a^{x}$

Cell	What varies / what's asked	Danger it hides
C0 Degenerate base $a = 1$	recognise $1^{x} = 1$	calling a flat line "exponential"
C1 Growth base $a > 1$	plot & read a steep rising curve	mixing up steepness with intercept
C2 Decay base $0 < a < 1$	falling curve, still $y > 0$	thinking output goes negative
C3 Negative exponent	evaluate $a^{- n}$	sign of the answer (still positive)
C4 Fractional exponent	evaluate $a^{p / q}$	reading a root off the graph
C5 Vertical shift $a^{x} + c$	new asymptote & intercept	leaving asymptote at $y = 0$
C6 Reflection / decay-as-mirror	$(1/ a)^{x}$ vs $a^{x}$	which axis we reflect in
C7 Solve for the exponent	find $x$ from $a^{x} = k$	needs a log — the inverse tool
C8 Limiting behaviour	$x \to \pm \infty$	which end goes to $0$
C9 Real-world word problem	build & use a model	units, "per hour" vs total
C10 Exam twist	combined shift + solve	asymptote and intercept and root

There are 11 cells (C0–C10) and 10 worked examples below. Example 0 handles the degenerate cell C0; Example 8 handles two cells (C8 and C6, cross-referenced there); the remaining eight take one cell each. Together they cover every cell.

Example 0 — Degenerate base · cell C0

Example 1 — Growth, base $a > 1$ · cell C1

Example 2 — Decay, base $0 < a < 1$ · cell C2

Worked example Sketch

$y = (\frac{1}{2})^{x}$ and confirm $y > 0$ Forecast: As $x$ grows, does the output ever dip below zero, or just get tiny?

Step 1. Rewrite $(\frac{1}{2})^{x} = 2^{- x}$ . Why this step? A base between $0$ and $1$ is easier to read as a negative exponent of a base $> 1$ — it flags "this is decay" immediately (see Index laws).

Step 2. Points: $x = 0 \Rightarrow 1$ , $x = 1 \Rightarrow \frac{1}{2}$ , $x = 2 \Rightarrow \frac{1}{4}$ , $x = - 1 \Rightarrow 2$ . Why this step? Right of $0$ it halves each step (shrinking, never zero); left of $0$ it doubles.

Step 3. Range stays $y > 0$ : a positive number to any power is positive.

In the figure, the yellow curve falls from left to right, and the pink dashed line marks the asymptote $y = 0$ . Notice the curve flattens toward that line on the right but never crosses it — every plotted dot sits strictly above $y = 0$ .

Verify: $(\frac{1}{2})^{2} = 0.25$ ✔, $(\frac{1}{2})^{- 1} = 2$ ✔, all outputs $> 0$ ✔.

Example 3 — Negative exponent evaluation · cell C3

Example 4 — Fractional exponent · cell C4

Example 5 — Vertical shift, moving the floor · cell C5

Worked example For

$y = 2^{x} - 5$ , state the asymptote, the $y$ -intercept, and the range Forecast: The bare $2^{x}$ had floor $y = 0$ . If we subtract $5$ , does the floor stay put?

Step 1. Subtracting $5$ lowers every output by $5$ , so the horizontal asymptote drops from $y = 0$ to $y = - 5$ . Why this step? An asymptote is a limiting height; shift the whole graph down and the limit shifts with it (see Graph transformations).

Step 2. $y$ -intercept: put $x = 0$ : $2^{0} - 5 = 1 - 5 = - 4$ , giving $(0, - 4)$ . Why this step? The intercept is just the function evaluated at $x = 0$ .

Step 3. Range: since $2^{x} > 0$ always, $2^{x} - 5 > - 5$ . So range is $y > - 5$ .

In the figure the pink dashed line has dropped to $y = - 5$ (that's the shift amount of $5$ below the old floor $y = 0$ ), and the blue dot marks the intercept $(0, - 4)$ .

Verify: As $x \to - \infty$ , $2^{x} \to 0$ , so $y \to - 5$ ✔ (approached, never reached). Intercept $2^{0} - 5 = - 4$ ✔.

Example 6 — Reflection: decay is growth in a mirror · cell C6

Worked example Show

$y = (\frac{1}{4})^{x}$ is $y = 4^{x}$ reflected in the $y$ -axis Forecast: Reflect in the $y$ -axis or the $x$ -axis?

Step 1. $(\frac{1}{4})^{x} = 4^{- x}$ . Why this step? Turning $\frac{1}{4}$ into $4^{- 1}$ exposes the hidden minus sign on the exponent.

Step 2. Replacing $x$ by $- x$ in any function reflects its graph across the $y$ -axis (it swaps left and right). Why this step? Left/right swap is a $y$ -axis mirror; up/down swap would be an $x$ -axis mirror. The minus is on $x$ , so it's the $y$ -axis.

In the figure the blue curve ( $4^{x}$ , growth) and the yellow curve ( $(1/4)^{x}$ , decay) are mirror images: the vertical $y$ -axis is the mirror line. Each point at height $h$ on the right of one curve appears at the same height $h$ on the left of the other, and both share the point $(0, 1)$ on the mirror.

Verify: At $x = 1$ : $(\frac{1}{4})^{1} = 0.25$ and $4^{- 1} = 0.25$ ✔. At $x = - 1$ : $(\frac{1}{4})^{- 1} = 4$ and $4^{1} = 4$ ✔ — reflected points.

Example 7 — Solve for the exponent (needs a log) · cell C7

Example 8 — Limiting behaviour, both ends · cells C8 + C6

Worked example Find

$x \to - \infty lim 3^{x}$ and $x \to + \infty lim (\frac{1}{3})^{x}$ Forecast: Which curve dies to $0$ on the left, and which on the right?

Step 1. Growth $3^{x}$ as $x \to - \infty$ . To make the far-left behaviour easy to read, change variable: let $N = - x$ , so that as $x$ marches to $- \infty$ , the new letter $N$ marches to $+ \infty$ (it's just the positive distance $x$ has travelled to the left). Substituting $x = - N$ gives $3^{x} = 3^{- N} = \frac{1}{3 ^{N}}$ . Since $N \to + \infty$ makes $3^{N} \to \infty$ , we get $\frac{1}{3 ^{N}} \to 0^{+}$ . Hence $lim_{x \to - \infty} 3^{x} = 0$ . Why this step? Introducing $N$ turns a scary "negative infinity exponent" into a plain "big positive $N$ " reciprocal — the parent's asymptote argument.

Step 2. Decay $(\frac{1}{3})^{x} = 3^{- x}$ . As $x \to + \infty$ , $- x \to - \infty$ , so by Step 1 this $\to 0^{+}$ . Hence $lim_{x \to + \infty} (\frac{1}{3})^{x} = 0$ . Why this step? This is cell C6 in action: decay dies at the right end precisely because it is growth mirrored in the $y$ -axis (Example 6). So this one example legitimately covers both cell C8 (limits) and cell C6 (reflection).

Verify: $3^{- 10} \approx 1.7 \times 1 0^{- 5} \approx 0$ ✔; $(\frac{1}{3})^{10} = 3^{- 10}$ same value $\approx 0$ ✔. Both $> 0$ , never exactly $0$ .

Example 9 — Real-world word problem · cell C9

Worked example A colony starts at

$200$ bacteria and triples every hour. How many after $4$ hours? After how many hours does it first exceed $50000$ ? Forecast: Is the " $200$ " a multiplier or a starting amount? Where does it sit in the formula?

Step 1. Build the model: after $x$ hours, $N (x) = 200 \cdot 3^{x}$ . Why this step? " $\times 3$ each hour" is exactly repeated multiplication — an exponential (see Exponential growth and decay models). The $200$ is the value at $x = 0$ , so it multiplies out front.

Step 2. After $4$ hours: $N (4) = 200 \cdot 3^{4} = 200 \cdot 81 = 16200$ . Why this step? Direct substitution — units are bacteria.

Step 3. Solve $200 \cdot 3^{x} = 50000 \Rightarrow 3^{x} = 250$ . This needs the inverse tool: $x = lo g_{3} 250 = \frac{ln 250}{ln 3} \approx 5.024$ hours. So it first exceeds $50000$ during the $6$ th hour — at whole hours, that's after $6$ hours ( $N (5) = 48600 < 50000 < N (6) = 145800$ ). Why a log? Same reason as Example 7 — only a logarithm frees $x$ from the exponent.

Verify: $N (4) = 200 \cdot 81 = 16200$ ✔. $N (5) = 200 \cdot 243 = 48600 < 50000$ ✔; $N (6) = 200 \cdot 729 = 145800 > 50000$ ✔ — so $6$ hours is the first whole hour above the threshold.

Example 10 — Exam twist: shift, intercept, root all at once · cell C10

Worked example The curve

$y = 2^{x} - 8$ meets the $x$ -axis at one point and has one asymptote. Find both, and the $y$ -intercept. Forecast: A shifted exponential can now genuinely cross $y = 0$ . Where?

Step 1. Asymptote: $2^{x} \to 0$ as $x \to - \infty$ , so $y \to - 8$ . Asymptote $y = - 8$ . Why this step? The $- 8$ drops the floor (cell C5). Note it's below the $x$ -axis, so a crossing is now possible.

Step 2. $x$ -axis crossing: set $y = 0$ : $2^{x} - 8 = 0 \Rightarrow 2^{x} = 8 = 2^{3} \Rightarrow x = 3$ . Root at $(3, 0)$ . Why this step? Matching bases gives the exponent directly — no log needed here since $8$ is a clean power.

Step 3. $y$ -intercept: $x = 0 \Rightarrow 2^{0} - 8 = 1 - 8 = - 7$ , giving $(0, - 7)$ .

In the figure the pink dashed asymptote sits at $y = - 8$ (a shift of $8$ below the old floor), the blue dot is the root $(3, 0)$ where the yellow curve pierces the $x$ -axis, and the off-white dot is the intercept $(0, - 7)$ .

Verify: $2^{3} - 8 = 0$ ✔ so $(3, 0)$ is on the axis. $2^{0} - 8 = - 7$ ✔. As $x \to - \infty$ , $y \to - 8$ ✔ (approached, not reached).

Active recall

Recall Match each prompt to its cell

What values may the base $a$ take, and why those? ::: $a > 0$ (else non-real roots like $a$ ) and $a \neq = 1$ (else $1^{x} = 1$ , a flat constant line). Cell C0 is the excluded $a = 1$ case. Solve $2^{x} = 20$ — which tool and why? ::: A logarithm $x = lo g_{2} 20$ ; only the log undoes the exponent (cell C7). Asymptote of $y = 2^{x} - 5$ ? ::: $y = - 5$ ; the shift lowers the floor (cell C5). Does $(\frac{1}{3})^{x}$ ever go negative? ::: No — positive base to any power is positive; range $y > 0$ (cell C2). Which end of $3^{x}$ dies to $0$ , and what substitution proves it? ::: The left, $x \to - \infty$ ; set $N = - x \to + \infty$ so $3^{x} = 1/ 3^{N} \to 0^{+}$ (cell C8). Why can $y = 2^{x} - 8$ cross the $x$ -axis but $y = 2^{x}$ cannot? ::: The shift drops the floor below $0$ , so a crossing exists at $x = 3$ (cell C10).

Connections

Logarithms as the inverse of exponentials — the tool that solves cells C7 and C9.
The number e and natural exponential eˣ — same behaviour with the natural base.
Index laws — every rewrite ( $a^{- n}$ , $a^{p / q}$ ) used above.
Exponential growth and decay models — cell C9 in the wild.
Graph transformations — shifts and reflections behind cells C5, C6, C10.

Worked examples — Exponential functions aˣ — graphs, properties, asymptote