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Exponentials (Growth, Decay, and SAT Shortcuts)

Exponentials: Growth, Decay, and SAT Shortcuts

Exponential problems look scary until you realize the SAT only tests a few ideas:
initial value, growth/decay factor, and long-run behavior.
If you can read those quickly, you win.

Big idea:
Exponentials are about multiplying by the same factor each step.
If the problem says “percent each year,” that’s your exponential signal.

Quick Jump (click to jump)
Everything you need for SAT exponentials, in one page
1) Recognize Exponentials Fast

Signals from words, tables, and graphs
 percentmultiplyratio
2) The Template: a(b)^t and a(1±r)^t

How to build models instantly
 initial valuefactor
3) Interpret Parameters (What the numbers mean)

“What does 120 represent?” type questions
 meaningunits
4) Long-Run Behavior + Horizontal Asymptotes

Why graphs approach a line
 asymptoteshift
5) Solving Exponential Equations (SAT-friendly)

Usually plug-and-chug or matching values
 substitutecompare
6) Common SAT Traps

Percent mistakes and wrong base
 trappercent
7) Practice

Model recognition + feature reading drills
 drillsspeed
8) SAT-Style Multiple Choice

Realistic questions + quick reasoning
 MCQsat style
9) Comparison Table (Cheat Sheet)

What to do based on the prompt
 fast lookuptest-day

Start: Recognize
Jump to MCQ
Jump to Cheat Sheet
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Recognize Exponentials Fast

If you can spot it, you’ll choose the right model immediately.

Word Signals

  • “increases by x% each year”
  • “decreases by x% per month”
  • “doubles every ___” / “halves every ___”
  • “compound interest” / “growth rate”
Rule: percent each step → multiply each step → exponential.

Table / Data Signals

  • If y-values multiply by the same factor → exponential
  • Check the ratio: y_{next}/y_{current}
50 → 60 → 72 → 86.4

Ratios: 1.2, 1.2, 1.2 ✅ exponential

Graph Signals

  • Growth curve: slow then fast upward
  • Decay curve: fast drop then flattens toward a line
  • Often has a horizontal asymptote (especially when shifted)

The Template: Build Exponential Models Instantly

Most SAT exponential models are one of these two forms.

Form 1: a(b)^t

f(t) = a(b)^t

a = initial value = f(0)

b = multiply factor each step
  • Growth: b > 1
  • Decay: 0 < b < 1

Form 2: a(1 ± r)^t

f(t) = a(1+r)^t (growth)

f(t) = a(1-r)^t (decay)
  • If rate is 12% growth → r = 0.12 → multiply by 1.12
  • If rate is 8% decay → r = 0.08 → multiply by 0.92

Mini Examples

Growth: starts at 300, increases 5% per year

P(t)=300(1.05)^t

Decay: starts at 80, decreases 20% per month

A(t)=80(0.80)^t

Interpret Parameters (What the Numbers Mean)

The SAT LOVES “what does this represent?” questions.

What does a mean?

f(t)=a(b)^t
  • a is the value when t=0
  • It’s the starting amount, initial population, initial price, etc.

What does b mean?

f(t)=a(b)^t
  • b is the factor each step
  • Percent change: if 15% growth → b=1.15; if 15% decay → b=0.85

Interpretation Example

g(t)=120(0.9)^t
  • 120 = starting value
  • 0.9 = multiply by 0.9 each step → 10% decrease per step

Time Unit Matters

“per year” vs “per month” changes what t means.
The SAT will try to trick you with mismatched units.

Check: Is t in years, months, days, or “number of periods”?

Long-Run Behavior + Horizontal Asymptotes

This is how the SAT tests “what happens as time goes on?”

Unshifted Decay Approaches 0

f(x)=a(b)^x where 0<b<1
  • As x→∞, b^x→0
  • So f(x)→0
  • Horizontal asymptote: y=0

Shifted Exponentials Approach y=k

f(x)=a(b)^x + k
  • As x→∞, a(b)^x→0 (for decay)
  • So f(x)→k
  • Horizontal asymptote: y=k
Example: 5(0.8)^x + 7 → asymptote y=7

Quick SAT rule:
If the function ends with “+ 6”, the asymptote usually becomes “y = 6”.

Solving Exponential Questions (SAT-Friendly)

The SAT rarely expects advanced log solving.
Most exponentials can be solved by substitution, matching values, or using answer choices.

Method 1: Plug in Small t

If you’re asked for a value after 1, 2, or 3 periods, just compute it directly.

P(t)=300(1.05)^t

P(2)=300(1.05)^2=300(1.1025)=330.75

Method 2: Use Ratios

From data, find b using consecutive values.

If y0=80 and y1=92,

b = 92/80 = 1.15

Method 3: Use Answer Choices (Backsolve)

If you’re solving for the rate or the factor, try choices quickly.

If model is A(t)=A0(1+r)^t,

test r values by checking if they match a given data point.

Common SAT Traps with Exponentials

  • Percent vs factor: 12% growth means multiply by 1.12 (not 0.12).
  • Decay confusion: 12% decrease means multiply by 0.88.
  • Units mismatch: “per month” but t is in years (or vice versa).
  • Linear trap: some students do 50 + 0.12t (that’s linear, not exponential).
  • Shifted asymptote trap: in a(b)^x + k, asymptote is y=k (not y=0).

Trap Example

“A value decreases by 25% each year.”
Wrong: multiply by 0.25
Right: multiply by 0.75

V(t)=V_0(0.75)^t

Practice: Exponentials

Train recognition + meaning + quick computation.

Set A: Build the Model

  1. Starts at 500 and increases 6% per year
  2. Starts at 120 and decreases 15% per month
  3. Doubles every hour starting from 40
  4. Halves every day starting from 96

Set B: Interpret the Parameters

  1. In P(t)=80(1.2)^t, what does 80 mean? What does 1.2 mean?
  2. In A(t)=300(0.85)^t, what percent change occurs each step?
  3. In f(x)=4(0.9)^x+6, what is the horizontal asymptote?

Set C: Quick Values

  1. If P(t)=200(1.1)^t, find P(1) and P(2)
  2. If A(t)=100(0.8)^t, find A(3)
  3. If f(x)=50(0.5)^x, find f(4)

Speed target:
You should be able to convert “x% per period” → multiplier in under 3 seconds.

SAT-Style Multiple Choice (Exponentials)

These reward correct modeling and feature reading.

Question 1

A quantity starts at 250 and increases by 12% each year. Which model represents the quantity after t years?

  1. 250 + 0.12t
  2. 250(1.12)^t
  3. 250(0.12)^t
  4. 250(1.88)^t

Question 2

For f(x)=80(0.9)^x, the value of f(0) is:

  1. 0.9
  2. 72
  3. 80
  4. 8

Question 3

The horizontal asymptote of g(x)=5(0.7)^x + 4 is:

  1. y=0
  2. y=4
  3. y=5
  4. x=4

Question 4

A population decreases by 20% each month. The population is 500 at month 0.
What is the population after 2 months?

  1. 300
  2. 320
  3. 400
  4. 450

Question 5

In the function h(t)=120(1.05)^t, what does 1.05 represent?

  1. The initial value
  2. The amount added each step
  3. The growth factor per time period
  4. The horizontal asymptote

Test-day move:
If the problem says “percent each period,” immediately write:
1 ± r as the multiplier. That alone eliminates most wrong choices.

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Cheat Sheet: Prompt → Model → First Move

Use this to decide what to do instantly.

If the prompt says… Your model is… First move
“increases by r% each period” A(t)=A0(1+r)^t Convert percent to decimal and add 1
“decreases by r% each period” A(t)=A0(1-r)^t Convert percent to decimal and subtract from 1
“doubles every k periods” A(t)=A0 \u22C5 2^{t/k} Each k steps multiplies by 2
“halves every k periods” A(t)=A0 \u22C5 (1/2)^{t/k} Each k steps multiplies by 1/2
“approaches a value of k” A(t)=A0(b)^t + k Horizontal asymptote is y=k
Data multiplies by constant factor A(t)=A0(b)^t Find b using ratio: next/current
Bottom line:
Exponentials become easy when you stop thinking “equations” and start thinking “multipliers.”

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