SAT-44: Exponential vs. Linear Growth

SAT-44: Exponential vs. Linear Growth

Description: The SAT loves to ask whether a situation is linear or exponential. The difference is simple: linear adds a fixed amount each step; exponential multiplies by a fixed factor each step.

The key distinction

• Linear: changes by a constant amount (repeated addition). Form: y = mx + b.
• Exponential: changes by a constant percent/factor (repeated multiplication). Form: y = a·b^x.

(Oʻzbekcha: chiziqli — har qadamda bir xil son qoʻshiladi; eksponensial — har qadamda bir xil songa koʻpaytiriladi.)

How to spot it from a table

Look at consecutive y-values:

• If you subtract and always get the same difference → linear.
• If you divide and always get the same ratio → exponential.

Tip: "doubles every…", "triples…", "grows 5% per…" all signal exponential. (Oʻzbekcha: "ikki barobar oshadi", "5% ga oshadi" — bular eksponensial belgilari.)

Worked Example

A table shows y-values 3, 6, 12, 24 for x = 0, 1, 2, 3. Linear or exponential?

• Differences: 3 → 6 is +3, 6 → 12 is +6. Not constant, so not linear.
• Ratios: 6/3 = 2, 12/6 = 2, 24/12 = 2. Constant ratio 2 → exponential, y = 3·2^x.

(Oʻzbekcha: nisbat doim 2 ga teng, demak eksponensial.)

Practice

Values 100, 80, 64, 51.2 — linear or exponential?

Key words — Kalit soʻzlar

• Linear growth — chiziqli oʻsish
• Exponential growth — eksponensial oʻsish
• Constant difference — oʻzgarmas ayirma
• Constant ratio — oʻzgarmas nisbat
• Add / Subtract — qoʻshish / ayirish
• Multiply / Divide — koʻpaytirish / boʻlish
• Decay — kamayish (so'nish)
• Factor — koʻpaytuvchi (koeffitsiyent)
• Rate — tezlik (sur'at)

Summary

• Linear = repeated addition (constant difference), y = mx + b.
• Exponential = repeated multiplication (constant ratio), y = a·b^x.
• From a table: equal differences → linear; equal ratios → exponential.