IB Maths AI HLFurther Functions & GraphsPaper 1 & 2Asymptote, growth, decay~8 min read
Exponential Functions & Graphs
An exponential y = k·ax + c (or y = kerx + 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 = kax + c or y = ka−x + c (with a > 0), and y = kerx + 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 = kax + c and the e-version y = kerx + 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 (ax, a−x or erx) 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 = −3e1.5x − 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 glancey = kax + c or y = kerx + cy-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
Asymptotey = 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 = 0y = 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 = −3e0.5x − 4.
read the constant outside the exponentialc = −4as x → −∞, e⁰·⁵ⁿ → 0, so y → −4asymptote y = −4k = −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 = −2e3x + 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 = cy = −12y-intercept: x = 0y = 3(1) − 12 = −9 ⇒ (0, −9)x-intercept: y = 03(2)ⁿ = 12 ⇒ 2ⁿ = 4 ⇒ x = 2range: k = 3 > 0 ⇒ above asymptotey > −12direction: 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 = cy = 3y-intercept: x = 0y = 4e⁰ + 3 = 4 + 3 = 7 ⇒ (0, 7)range: k = 4 > 0 ⇒ above asymptotey > 3x-intercept? set y = 04e⁻²ⁿ = −3 ⇒ impossible (eⁿ > 0)no x-interceptdirection: k = + (4), exp coef = − (−2) ⇒ differenty = 3 · (0, 7) · range y > 3 · no root · decreasingwhole 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) = 200e0.15t, 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 = 200initial population: 200(b) N(10)N(10) = 200e¹·⁵ ≈ 200(4.4817)N(10) ≈ 896.34(c) solve 200e⁰·¹⁵ⁿ = 1000e⁰·¹⁵ⁿ = 50.15t = ln 5 ⇒ t = ln 5 / 0.15t ≈ 10.73 hoursno 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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