SAT-45: Writing Exponential Functions (y = ab^x) from Word Problems

SAT-45: Writing Exponential Functions (y = ab^x) from Word Problems

Description: Once you know a situation is exponential (SAT-44), you write it as y = a·b^x. This lesson shows how to pull a and b out of the words.

What a and b mean

• a = the starting value (when x = 0).
• b = the growth/decay factor per step.

Build b from the percent rate:

• Growth of r% → b = 1 + r/100 (e.g. +8% → b = 1.08).
• Decay of r% → b = 1 − r/100 (e.g. −8% → b = 0.92).

(Oʻzbekcha: a — boshlangʻich qiymat, b — har bir qadamdagi oʻzgarish koeffitsiyenti.)

Worked Example (growth)

A town has 5,000 people and grows 3% per year. Write the population after x years.

• Starting value a = 5000.
• Growth 3% → b = 1 + 0.03 = 1.03.
• Function: y = 5000·(1.03)^x.

(Oʻzbekcha: 3% oʻsish uchun b = 1.03.)

Worked Example (decay)

A car worth $20,000 loses 15% of its value each year.

• a = 20000, decay 15% → b = 1 − 0.15 = 0.85.
• Function: y = 20000·(0.85)^x.

Tip: "doubles" → b = 2; "halves" → b = 0.5; "triples" → b = 3. (Oʻzbekcha: "ikki barobar" → b = 2; "yarmiga" → b = 0.5.)

Practice

A colony of 200 bacteria doubles every hour. Write y for hours x.

Key words — Kalit soʻzlar

• Exponential function — eksponensial funksiya
• Starting value (initial) — boshlangʻich qiymat
• Growth factor — oʻsish koeffitsiyenti
• Decay factor — kamayish koeffitsiyenti
• Rate (percent) — foiz sur'ati
• Per year / per hour — har yili / har soatda
• Double / Halve — ikki barobar / yarmiga
• Base — asos
• Exponent — daraja koʻrsatkichi

Summary

• Model: y = a·b^x; a = starting value, b = factor per step.
• Growth r% → b = 1 + r/100; decay r% → b = 1 − r/100.
• "Doubles" → b = 2, "halves" → b = 0.5, etc.