Exponential growth: graph the model, tap the point you need

An original SAT-style practice question on exponential growth, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with the by-hand method (and logarithms, for when the numbers aren't as tidy) alongside.

Practice question

A biology class tracks a bacteria culture. The number of bacteria, t hours after the class starts observing, is modeled by N(t) = 600 · 2^t/4. According to the model, how many hours after the observation starts will the culture contain 19,200 bacteria?

⚡ Desmos is the fastest way here Graph the model and the target as two lines, then tap where they cross — no dividing, no powers of 2, no logarithms.

Exponential functionsGrowth modelsMedium⏱ ~40 sec

Step by step

Graph the model, using x for t, and the target value

Type the model on one line and the number of bacteria you're looking for on the next. After the ^, Desmos keeps typing in the exponent until you press the right arrow, so check that the whole x/4 sits up top. The y-values here run into the tens of thousands, so zoom out, or open Graph Settings (the wrench) and set the y-axis to about 25,000 and the x-axis to about 25.

y=600(2)^{x/4}Blue curve, rising faster and faster

y=19200Red horizontal line

What you see: a curve that starts low on the left and climbs steeply, crossing a flat line high up on the right.

Tap the intersection

Tap where the curve meets the horizontal line. Desmos drops a grey point there and labels its coordinates.

(20,19200)Where the curve reaches 19,200

What you see: a labelled point at (20, 19200). The y-value just repeats the target; the x-value is the time.

Read off the coordinate the question asked for

The question asks for the number of hours, which is the x-coordinate.

20Hours — done

Answer

20 hours

The culture reaches 19,200 bacteria 20 hours after the observation starts. Check: 600 · 2^20/4 = 600 · 2⁵ = 600 · 32 = 19,200 ✓.

⚠ Common trap

Answering 5 — that's the number of doublings (2⁵ = 32), not the number of hours. Each doubling takes 4 hours, so the time is 5 × 4 = 20. A second slip by hand: multiplying 600 × 2 first. The exponent applies only to the 2, so the model is not 1200^t/4.

The algebra way, if you prefer

Divide, then match the powers: set 600 · 2^t/4 = 19,200. Divide both sides by 600: 2^t/4 = 32. Since 32 = 2⁵, the exponents must match: t/4 = 5, so t = 20.

With logarithms, for when the ratio isn't a neat power of 2: t/4 = log₂(32), or t = 4 · log(32) / log(2) = 4 · 5 = 20. Here the numbers are friendly, so the by-hand route is quick. When the target is something like 15,000, the graph gives you the answer without any logs.

More walkthroughs: line of best fit · vertex and maximum value · quadratic roots · systems of linear equations · circle equations · sum of a quadratic's solutions · solving for x by graphing · linear inequalities · absolute value equations · Desmos cheat sheet · all walkthroughs

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