IB Maths AI SL Topic 2 — Further Functions & Graphs Paper 1 & 2 Growth & decay models ~7 min read

Exponential Functions & Graphs

Exponential curves break the polynomial pattern. Instead of crossing the x-axis multiple times, they approach a horizontal asymptote on one side and shoot off to infinity on the other. The general form is y = k·a^x + c (or y = ke^rx + c using the special base e ≈ 2.718). Three numbers shape the curve: c sets the asymptote, k decides above/below it, and the sign of the exponent decides growth or decay.

📘 What you need to know

Standard form: y = k·a^x + c with a > 0, a ≠ 1 — or the natural form y = ke^rx + c, where e ≈ 2.718.
Horizontal asymptote: y = c — read it straight from the equation. The curve approaches but never touches this line.
y-intercept: at x = 0 the a^x (or e^rx) term equals 1, so y = k + c.
Side of asymptote: k > 0 ⇒ curve sits ABOVE asymptote (range y > c); k < 0 ⇒ BELOW (range y < c).
Growth or decay: if k and the exponent’s coefficient have the SAME sign, the curve is increasing; if OPPOSITE signs, decreasing. For base < 1 the rule flips.
At most 1 root: an exponential crosses the x-axis at most once — only if k and c have opposite signs. Use the GDC to find it.

The asymptote and the y-intercept

Every exponential of the form y = k·a^x + c has the line y = c as its horizontal asymptote — that’s where the exponential term tends to zero as x heads to one of the infinities. The y-intercept is reached by putting x = 0 into the equation: since a⁰ = 1, you always get y = k + c.

Exponential — key landmarks y = k·a^x + c or y = ke^rx + c

asymptote y = c · y-intercept (0, k + c) · range depends on sign of k

The dashed red line is the horizontal asymptote y = 1. The curve sits above it (since k = 1 > 0), passes through the y-intercept at (0, 2), and climbs steeply to the right.

Growth vs decay — which way does it go?

An exponential curve either grows (rises to the right) or decays (falls to the right). Two things decide which: the sign of k (above or below asymptote) and the sign of the exponent’s coefficient (or whether the base is > 1 or < 1). The cleanest mental check is to evaluate at one large x and one small x: if y grows with x, it’s increasing.

Quick rules: for y = ke^rx + c — if k and r have the SAME sign, the curve is INCREASING; if OPPOSITE signs, DECREASING. For y = k·a^x + c with k > 0: base > 1 grows, base < 1 decays.

Real-world models

Exponential functions model continuous growth or decay: bacteria multiplying, radioactive material decaying, investments compounding, hot drinks cooling. The base a (or rate r) sets the speed; k sets the initial offset above the asymptote; c is the long-term limit. In a cooling problem, c is the room temperature — the object never gets cooler than its surroundings.

🧭 Recipe — sketching any exponential

Read off the asymptote: it’s the constant c at the end. Draw it as a dashed line y = c.
Find the y-intercept: substitute x = 0 (or just compute k + c). Plot (0, k + c).
Decide which side of the asymptote: k > 0 ⇒ curve above; k < 0 ⇒ below. Decide growth/decay from the signs.
Plot one extra point: a value at x = 1 or x = −1 anchors the curve’s steepness.
Draw the curve: smooth, approaching the dashed asymptote on one side and shooting off to ±∞ on the other. Label the asymptote with its equation.

Worked examples

WE 1

Standard exponential — find all features

For the function y = 3(2)^x − 5, state the equation of the horizontal asymptote, find the y-intercept and the x-intercept (to 3 s.f.), and decide whether the curve is increasing or decreasing.

Step 1 — identify k, base, c k = 3, base = 2, c = −5 Step 2 — asymptote = c y = −5 Step 3 — y-intercept (x = 0) y = 3(1) − 5 = −2 → (0, −2) Step 4 — x-intercept (GDC solver: 3(2)ˣ − 5 = 0) x ≈ 0.737 Step 5 — growth or decay? k > 0 (above asymptote) and base 2 > 1 ⇒ INCREASING curve asymptote y = −5 · y-int (0, −2) · x-int (0.737, 0) · increasing k = 3 > 0 and c = −5 have OPPOSITE signs, so the curve must cross the x-axis exactly once. That’s why this exponential has a root.

WE 2

Negative coefficient — reflected exponential

The function f(x) = −4e^x + 6 is defined for all real x. Find the asymptote, the y-intercept and the x-intercept (to 3 s.f.). State the range.

Step 1 — identify k, r, c k = −4, r = 1, c = 6 Step 2 — asymptote y = 6 Step 3 — y-intercept f(0) = −4(1) + 6 = 2 → (0, 2) Step 4 — x-intercept (GDC) −4eˣ + 6 = 0 ⇒ eˣ = 1.5 x ≈ 0.405 Step 5 — range (k < 0 ⇒ BELOW asymptote) y < 6 asymptote y = 6 · y-int (0, 2) · x-int (0.405, 0) · range y < 6 negative k flips the curve to sit BELOW the asymptote. Same-sign k and r would mean increasing; here k and r differ, so the curve DECREASES (heads to −∞ as x → +∞).

WE 3

Radioactive decay

A radioactive substance has mass (mg) at time t (years) modelled by m(t) = 50e^−0.02t. Find the initial mass, the mass after 30 years (to 3 s.f.), and the long-term behaviour.

Step 1 — initial mass (t = 0) m(0) = 50e⁰ = 50 mg Step 2 — mass after 30 years m(30) = 50e⁻⁰‧⁶ = 50 × 0.5488… ≈ 27.4 mg Step 3 — long-term: as t → ∞, e⁻⁰‧⁰²ᵗ → 0 m → 50(0) = 0 asymptote m = 0 (substance never quite disappears) m(0) = 50 mg · m(30) ≈ 27.4 mg · asymptote m = 0 classic decay: c = 0 means the asymptote is the t-axis itself. After 30 years the mass is about 55% of the original — the formula encodes the decay rate.

WE 4

Decay with base < 1 and a non-zero asymptote

For the function y = 10(0.5)^x + 2, find the y-intercept and the asymptote, decide whether the curve is increasing or decreasing, and state the range.

Step 1 — identify k, base, c k = 10, base = 0.5, c = 2 Step 2 — asymptote y = 2 Step 3 — y-intercept y(0) = 10(1) + 2 = 12 → (0, 12) Step 4 — growth or decay? base 0.5 < 1 with k > 0 ⇒ DECREASING Step 5 — range k > 0 ⇒ curve ABOVE asymptote y > 2 (never reaches 2) asymptote y = 2 · y-int (0, 12) · decreasing · range y > 2 no x-intercept here: k and c are both positive, so the curve sits entirely above the x-axis. The curve decays from (0, 12) towards the asymptote y = 2.

WE 5

Find the equation from features

An exponential function of the form y = k·a^x + c has horizontal asymptote y = 1, passes through (0, 7), and also passes through (2, 25). Find the values of k, a and c.

Step 1 — asymptote gives c c = 1 Step 2 — use (0, 7) to find k 7 = k(a⁰) + 1 = k + 1 k = 6 Step 3 — use (2, 25) to find a 25 = 6(a²) + 1 6a² = 24 a² = 4 ⇒ a = 2 (positive base) y = 6(2)ˣ + 1 three unknowns (k, a, c) need three independent facts: the asymptote pins c, then each remaining point pins k and a in turn. Standard 3-equations setup.

WE 6

Investment doubling time (GDC)

An investment of $5000 grows continuously at 4.5% per year, modelled by A(t) = 5000e^0.045t. Find the value after 10 years (to the nearest dollar), and use a graphical method to find how long until the investment doubles in value.

Step 1 — value after 10 years A(10) = 5000e⁰‧⁴⁵ = 5000 × 1.5683… ≈ $7842 Step 2 — doubling: solve A(t) = 10000 graphically plot Y₁ = 5000e⁰‧⁰⁴⁵ᵗ and Y₂ = 10000 use intersect tool t ≈ 15.4 years Step 3 — verify A(15.4) ≈ 5000 × e⁰‧⁶⁹³ ≈ 10000 ✓ A(10) ≈ $7842 · doubles after ≈ 15.4 years graphical method works without logs (which come in a later note). “Continuous” growth uses e, distinguishing it from yearly compounding which uses (1 + r)ᵗ.

💡 Top tips

Asymptote = the constant: for any y = k·a^x + c, the asymptote is just y = c. Read it off, don’t compute it.
y-intercept shortcut: at x = 0, a⁰ = 1, so the intercept is always k + c. No calculator needed.
For sketches, ALWAYS draw the asymptote as a dashed line: examiners check for this. Forgotten asymptotes cost marks even if the curve is correct.
Check growth vs decay with one extra point: if y(1) > y(0), growing; otherwise decaying. Sign-tracking rules are nice, but a single check on the GDC is foolproof.
“At most one root”: an exponential curve can cross the x-axis 0 or 1 times. If k and c have opposite signs, there’s one root; otherwise none.

⚠ Common mistakes

Setting the asymptote to y = 0 by default: it’s y = c, the constant. For y = 2^x + 3 the asymptote is y = 3, not y = 0.
Forgetting to add the constant for the y-intercept: y = 5e^x − 2 has y-int 5 + (−2) = 3, not just 5.
Drawing the curve crossing the asymptote: it never does. The curve gets arbitrarily close but never touches.
Confusing the inverse with the reciprocal: e^−x is NOT 1/e — it equals 1/e^x. The minus sign in the exponent flips a growth curve into a decay curve.
Mixing up the GDC mode: make sure the calculator is in radian/numerical mode (not degree) when using e. e^x doesn’t care about angle modes, but other GDC operations can.

Up next: Sinusoidal Functions & Graphs. Sine and cosine bring something new — periodic behaviour. Instead of one global shape, sinusoids repeat forever along the x-axis. The four key features (amplitude, period, principal axis, intercepts) replace turning points and asymptotes, and they’re all readable straight from the equation y = a sin(bx) + d.

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