Desmos for the SAT · Technique 9 of 23

Model Exponential Growth and Decay

Graph y = a·b^x and read the starting value, the trend, and any future value.

SolvesExponential growthExponential decayGrowth and decay modelsValue at a time

Watch the lesson

The whole move worked on one real SAT question, narrated.

0:00 / 0:00

The same question, worked in the calculator: type the model, read day 0 off the y-axis, then give day 3 its own line and click the crossing.

Exponential questions on the SAT hand you a model like $y = a \cdot b^{x}$ and ask for the starting amount, whether it grows or shrinks, or the value after some time. All three are sitting right on the graph.

Two numbers do all the work: a is the starting value (where the curve meets the y-axis at $x = 0$ ), and b is the factor it multiplies by each step. If b is bigger than 1 the graph grows; if b is between 0 and 1 it decays.

Type the model in exactly as written, using ^ for the exponent, and Desmos draws the whole story. For the value at a particular time, type that x as its own line and click the crossing, or define the model as a function and ask for f(3) directly. No power arithmetic by hand.

Nova

An exponential model gives you everything on the graph: where it starts, whether it is growing or shrinking, and its value at any time. Read the picture instead of working the powers by hand.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps raising the base to a power by hand.

The question

$y = 50(3)^{x}$ , a colony that starts with 50 cells and triples every day

What is the population after 3 days?

The algebra way

Plug the time in: 50 · 3^3.
Work the power, then multiply: 3^3 = 27, and 50 · 27 = 1350.
Stop one power early and the result is 450, choice B. Go one too far and it is 4050, choice D.

The Desmos way

Type y = 50(3)^x, then x = 3 on the next line, and click the crossing: (3, 1350).
Or define f(x) = 50(3)^x and type f(3). Desmos prints 1350.

The payoff: The value on any day is a click or a one-line function call, so there is no power to work by hand and no day to miscount.

Quick reference

Reading $y = a \cdot b^{x}$

Part	What it tells you
a (the y-intercept)	The starting value, at $x = 0$
b greater than 1	Growth, the curve rises
b between 0 and 1	Decay, the curve falls
$b = 1 + r$	Grows by rate r each step, e.g. 1.2 is 20% growth

When the exponent has a period in it

The model	What one period means	The percent
$f(x) = 5470(0.64)^{x/12}$ , x in months	The exponent reaches 1 when $x = 12$ , so one period is a year	Multiplied by 0.64 each year, a 36% decrease
$P(t) = 260(1.04)^{1.5t}$ , t in years	The exponent reaches 1 when $t = 2/3$ , so one period is 8 months	Multiplied by 1.04 each period, a 4% increase
Any $a \cdot b^{x/k}$	One full factor of b takes k steps of x	b is the factor per k steps, so its percent is per k steps too

Step by step

The whole move, all 5 steps

Type the model exactly as given, using ^ for the exponent, then press the right arrow to get back out of the exponent.
Read the starting value where the curve crosses the y-axis, at $x = 0$ .
Decide growth or decay from the shape: a base above 1 rises, a base below 1 falls.
For the value at a particular time, type that x as its own line, x = 3, click inside either equation box to reveal the gray dots, then click the crossing. Or define the model as f(x) = ... and type f(3) on the next line to print the value.
Want a percent per period? Evaluate the model one period apart and divide: f(12)/f(0) gives the factor, and the percent is how far that factor sits from 1.

The question

A bacteria colony starts with 50 cells and triples in number every day. The population is modeled by y = 50(3)^x, where x is the number of days.

What is the population after 3 days?

A150
B450
C1350
D4050

Step 1 of 4

Type the model exactly as the question writes it, y = 50(3)^x, then press Enter. The x is the only thing in the exponent, so nothing has to step back out of it before the line ends.

In the calculator: y=50(3)^x

Parentheses around the base keep the 50 and the 3 apart, so Desmos reads it as 50 times 3 to the x and not 503 to something.

Syntax: Type the base in parentheses and the exponent with ^: 50(3)^x. To type anything after the exponent, press the right arrow first. The y-intercept (the value at $x = 0$ ) is the starting amount a; a base above 1 grows, a base between 0 and 1 decays.

Now try it on a real SAT question

Head or scratch paper. No graph needed: a base below 1 falls, a base above 1 rises. Scan the four bases and one of them is under 1. Graph all four afterwards if you want to see it, but the reading is faster.

Four exponential models are given.

Which model shows exponential decay?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

When the exponent is divided, like $b^{x/12}$ , the base is the factor per 12 steps of x, not per one step. Check which period the question asks about before turning b into a percent; f(1)/f(0) gives the per-step factor if that is the one it wants.
If the question only wants the starting value or whether it grows or decays, read a and b straight from the equation. No graph needed.
For "how long until it reaches N", graph the model and the line $y = N$ and find where they cross. That is the solve-by-graphing move.
Watch compound-rate wording: y = a(1 + r)^t uses the rate r, so the base is (1 + r). A 20% rate means a base of 1.2, not 0.2.

Nova

If the question only asks for the starting value or growth versus decay, read a and b straight off the equation. That's faster than opening the calculator at all.

All Desmos techniques

Put this move into real practice

"Model Exponential Growth and Decay" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.