IB Maths AI HL Further Functions & Graphs Paper 1 & 2 Asymptote, growth, decay ~8 min read

Exponential Functions & Graphs

An exponential y = k·a^x + c (or y = ke^rx + c) has a single horizontal asymptote y = c, a y-intercept of (0, k + c), and either rises or falls depending on whether the signs of k and the exponent coefficient agree.

📘 What you need to know

Forms: y = ka^x + c or y = ka^−x + c (with a > 0), and y = ke^rx + c with e ≈ 2.718.
y-intercept: (0, k + c) — substitute x = 0.
Horizontal asymptote: y = c. The graph hugs this line but never crosses it.
Range: if k > 0 the graph sits above the asymptote, so y > c. If k < 0 it sits below, so y < c.
Increasing/decreasing: signs of k and the exponent coefficient agree ⇒ increasing; disagree ⇒ decreasing.
At most one x-intercept; sometimes none (when the curve and the asymptote are on the same side of the x-axis).

The standard form and its features

Both y = ka^x + c and the e-version y = ke^rx + c behave the same way: the constant c shifts the whole curve vertically and becomes the horizontal asymptote; the coefficient k stretches it and chooses which side of the asymptote it sits on; the exponent rule (a^x, a^−x or e^rx) decides which direction it grows. Setting x = 0 gives the y-intercept directly: k·1 + c = k + c.

Increasing vs decreasing — reading the signs

An exponential graph is either monotonically increasing or monotonically decreasing — it never turns. The simple rule: compare the sign of k with the sign of the exponent’s coefficient (the x coefficient inside the power). Same sign ⇒ increasing; different signs ⇒ decreasing. So y = 3(2)^x + 1 and y = −3e^−1.5x + 4 both increase, while y = 3(2)^−x + 1 and y = −3e^1.5x − 4 both decrease.

Both graphs share y = c as their horizontal asymptote and (0, k + c) as their y-intercept; only the direction differs.

Exponential function at a glance y = ka^x + c or y = ke^rx + c y-intercept (0, k + c) · asymptote y = c · range y > c (if k > 0) or y < c (if k < 0)

Sketching an exponential

The features come in a fixed order. Read off c — that’s the asymptote. Substitute x = 0 for the y-intercept (k + c). Check the sign of k to see which side of the asymptote the curve sits on; check the signs of k and the exponent coefficient to determine direction. If the question asks for an x-intercept, set y = 0 and solve — there’s at most one, and sometimes none.

Asymptotes aren’t drawn by the GDC — read the equation. A constant added to the exponential part is the asymptote; for y = 4e^−2x + 3 it’s y = 3.

🧭 Recipe — sketching an exponential

Asymptote y = c — the constant term outside the exponential.
y-intercept at (0, k + c) — substitute x = 0.
Side of the asymptote: above if k > 0, below if k < 0.
Direction: signs of k and the exponent coefficient agree ⇒ increasing; disagree ⇒ decreasing.
x-intercept (if asked): solve y = 0 on the GDC; if the asymptote and the y-intercept are on the same side of the x-axis, there isn’t one.

Worked examples

WE 1

y-intercept of an exponential

Find the y-intercept of y = 2(5)^x + 3.

substitute x = 0 y = 2(5)⁰ + 3 = 2(1) + 3 = 5 (0, 5) shortcut: the y-intercept is always k + c ⇒ 2 + 3 = 5.

WE 2

Asymptote of a translated exponential

State the horizontal asymptote of y = −3e^0.5x − 4.

read the constant outside the exponential c = −4 as x → −∞, e⁰·⁵ⁿ → 0, so y → −4 asymptote y = −4 k = −3 < 0, so the curve sits below the asymptote.

WE 3

Increasing or decreasing?

Classify each function as increasing or decreasing: (a) y = 3(2)^x + 1; (b) y = 3(2)^−x + 1; (c) y = −2e^3x + 5; (d) y = −4(2)^−x + 1.

compare signs of k and the exponent coefficient (a) k = 3 (+), exp coef = 1 (+) ⇒ same ⇒ increasing (b) k = 3 (+), exp coef = −1 (−) ⇒ different ⇒ decreasing (c) k = −2 (−), exp coef = 3 (+) ⇒ different ⇒ decreasing (d) k = −4 (−), exp coef = −1 (−) ⇒ same ⇒ increasing (a) inc · (b) dec · (c) dec · (d) inc

WE 4

Full feature list (with x-intercept)

For y = 3(2)^x − 12, find the asymptote, y-intercept, x-intercept, range and direction.

asymptote: y = c y = −12 y-intercept: x = 0 y = 3(1) − 12 = −9 ⇒ (0, −9) x-intercept: y = 0 3(2)ⁿ = 12 ⇒ 2ⁿ = 4 ⇒ x = 2 range: k = 3 > 0 ⇒ above asymptote y > −12 direction: both signs + y = −12 · (0, −9) · (2, 0) · range y > −12 · increasing

WE 5

Full feature list (no x-intercept)

For y = 4e^−2x + 3, find the asymptote, y-intercept, range and direction. Does it have an x-intercept?

asymptote: y = c y = 3 y-intercept: x = 0 y = 4e⁰ + 3 = 4 + 3 = 7 ⇒ (0, 7) range: k = 4 > 0 ⇒ above asymptote y > 3 x-intercept? set y = 0 4e⁻²ⁿ = −3 ⇒ impossible (eⁿ > 0) no x-intercept direction: k = + (4), exp coef = − (−2) ⇒ different y = 3 · (0, 7) · range y > 3 · no root · decreasing whole curve sits above y = 3, so it can never reach the x-axis.

WE 6

Applied: bacterial growth

A bacterial colony has size N(t) = 200e^0.15t, where t is in hours. (a) Find the initial population. (b) Find N(10). (c) Find the time at which the population reaches 1000 (to 2 d.p.).

(a) N(0) N(0) = 200e⁰ = 200 initial population: 200 (b) N(10) N(10) = 200e¹·⁵ ≈ 200(4.4817) N(10) ≈ 896.34 (c) solve 200e⁰·¹⁵ⁿ = 1000 e⁰·¹⁵ⁿ = 5 0.15t = ln 5 ⇒ t = ln 5 / 0.15 t ≈ 10.73 hours no asymptote shown by GDC: read it from the equation (here c = 0, so y = 0).

💡 Top tips

y-intercept shortcut: it’s always (0, k + c) — no substitution needed.
The asymptote is the constant term outside the exponential, never the base.
Sign-pair rule for direction: same signs of k and the exponent coefficient ⇒ increases; otherwise decreases.
If the curve’s y-intercept and asymptote are on the same side of the x-axis, there’s no x-intercept.
For exponential applications, set the model equal to the target value and use the GDC’s intersect (or ln) to find the time.

⚠ Common mistakes

Drawing the curve through the asymptote — by definition it can never cross.
Reading the base as the asymptote — the base controls steepness, not the asymptote.
Saying “no roots” by default when there actually is one (e.g. when k and the asymptote disagree in sign).
Sign slip in the direction rule: missing a negative inside the exponent flips the answer.
Forgetting the asymptote exists because the GDC doesn’t draw it.

Next up: Sinusoidal Functions & Graphs — sine and cosine curves with amplitude, period, principal axis and phase shift.

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