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Digital SAT · Solve it in Desmos

How to Solve Exponential Growth and Decay on Desmos (Digital SAT)

Use Desmos to confirm which equation fits by checking key points, or to evaluate an amount after a given time. You build the model from the starting value and factor first; the graph confirms it.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for exponential growth and decay

The Desmos steps

  1. 1

    Graph the candidate

    Type the equation you suspect as y = ... to see its curve.

  2. 2

    Check the key points

    Confirm the y-intercept is the starting value and a later point matches the description.

  3. 3

    Compare choices

    Graph each option and keep the one whose start and growth fit; wrong factors miss the points.

Worked in Desmos

Easy example

An investment of 500500 dollars grows by 3%3\% each year.

Which equation gives the value yy, in dollars, after xx years?
y=500(1.03)xy = 500(1.03)^{x}
B
y=500(0.03)xy = 500(0.03)^{x}
C
y=500(0.97)xy = 500(0.97)^{x}
D
y=500(1.3)xy = 500(1.3)^{x}

A: Correct. Growth of 3%3\% means a factor of 1+0.03=1.031 + 0.03 = 1.03, so y=500(1.03)xy = 500(1.03)^{x}.

B: Incorrect. This uses the rate 0.03 as the factor instead of 1+0.031 + 0.03.

C: Incorrect. This is a decay factor 1−0.031 - 0.03; the value grows, not shrinks.

D: Incorrect. This misplaces the decimal, using 1.31.3 (a 30%30\% increase) instead of 1.031.03.

Explanation

A 3%3\% yearly increase gives a growth factor of 1.031.03, so y=500(1.03)xy = 500(1.03)^{x}.

Medium example

A 60-milligram sample loses one-third of its mass each hour.

Which equation gives the mass mm, in milligrams, remaining after hh hours?
A
m=60(1.33)hm = 60(1.33)^{h}
B
m=60(13)hm = 60\left(\frac{1}{3}\right)^{h}
C
m=60−13hm = 60 - \frac{1}{3}h
m=60(23)hm = 60\left(\frac{2}{3}\right)^{h}

A: Incorrect. A factor above 1 models growth, not a loss.

B: Incorrect. A factor of 13\frac{1}{3} removes two-thirds each hour, not one-third.

C: Incorrect. This is linear decay, not a one-third proportional loss.

D: Correct. Losing one-third leaves two-thirds, so the factor is 23\frac{2}{3}.

Explanation

Retaining 23\frac{2}{3} each hour: m=60(23)hm = 60\left(\frac{2}{3}\right)^{h}.

Hard example

A video has 500 views, and the number of views triples each day.

How many views will the video have after 3 days?
A
15001500
B
35003500
C
45004500
1350013500

A: Incorrect. This is the count after only 1 day: 500(3)500(3).

B: Incorrect. This treats growth as linear: 500+3000500 + 3000.

C: Incorrect. This is the count after only 2 days: 500(3)2500(3)^{2}.

D: Correct. 500(3)3=500(27)=13500500(3)^{3} = 500(27) = 13500.

Explanation

500(3)3=13500500(3)^{3} = 13500.

Detailed explanation

Tripling each day for 3 days multiplies by 33=273^{3} = 27, so 500×27=13,500500 \times 27 = 13{,}500. Stopping after 1 or 2 days, or adding instead of multiplying, gives the smaller values.

Try it with the calculator

Question 1easy

A machine is worth 800800 dollars and loses 10%10\% of its value each year.

Which equation gives the machine's value yy, in dollars, after xx years?

Question 2easy

A culture begins with 12001200 bacteria, and the number triples every hour.

Which equation gives the number of bacteria yy after xx hours?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Desmos questions

Can Desmos solve exponential growth and decay questions on the SAT?

Use Desmos to confirm which equation fits by checking key points, or to evaluate an amount after a given time. You build the model from the starting value and factor first; the graph confirms it.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master exponential growth and decay, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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