SAT-65: Linear vs. Exponential Data Modeling

SAT-65: Linear vs. Exponential Data Modeling

Description: This lesson ties together SAT-44 and SAT-54: given a table, graph, or description, decide whether a linear model or an exponential model fits, and write it. Picking the wrong model is a common SAT mistake, so the decision rule matters.

The decision rule

• Change by a constant amount each step → linear, y = mx + b.
• Change by a constant percent/factor each step → exponential, y = a·b^x.

(Oʻzbekcha: bir xil son qoʻshilsa — chiziqli; bir xil foizga oʻzgarsa — eksponensial.)

Diagnosing from a table

Take the y-values in order. Subtract neighbors: if the differences are equal, it's linear. Divide neighbors: if the ratios are equal, it's exponential. Do both checks before deciding. (Oʻzbekcha: qoʻshni qiymatlarni ayiring (chiziqli?) va boʻling (eksponensial?).)

Diagnosing from a graph

• A straight line → linear.
• A curve that grows faster and faster (or decays toward zero without crossing) → exponential.

Worked Example 1 — table that is linear

x: 0, 1, 2, 3 and y: 5, 8, 11, 14. Which model, and write it.

• Differences: +3, +3, +3 → constant → linear.
• Slope m = 3, starting value b = 5 → y = 3x + 5.

Worked Example 2 — table that is exponential

x: 0, 1, 2, 3 and y: 4, 12, 36, 108. Which model, and write it.

• Differences: +8, +24, +72 (not constant). Ratios: 12/4 = 3, 36/12 = 3, 108/36 = 3 → constant → exponential.
• Start a = 4, factor b = 3 → y = 4·3^x.

(Oʻzbekcha: nisbat doim 3 ga teng, demak eksponensial: y = 4·3^x.)

Worked Example 3 — from a description

"A pond's algae covers 50 m² and increases 20% each week." Linear or exponential? Write it.

• "Increases 20% each week" = constant percent → exponential.
• a = 50, b = 1 + 0.20 = 1.2 → y = 50·(1.2)^x.

Watch the words: "per year/week" with a fixed amount → linear; with a fixed percent → exponential. (Oʻzbekcha: aniq son qoʻshilsa — chiziqli; foiz boʻlsa — eksponensial.)

Why the models look so different far out

Near the start, a linear and an exponential model can look almost the same — both rise gently. The difference shows up later: a linear model keeps adding the same step forever, so its graph is a straight line, while an exponential model keeps multiplying, so it eventually shoots up far steeper (or, for decay, flattens toward zero without ever touching it). This is why "which model fits long-term growth like population or money?" is almost always exponential, and why a constant hourly wage or a fixed monthly fee is linear. (Oʻzbekcha: boshida ikkala model oʻxshash koʻrinadi, lekin keyinroq eksponensial ancha tikroq oʻsadi.)

Watch out for "starting value" wording

The constant term carries different names in word problems: "starting amount," "initial value," "flat fee," or "value when x = 0." In a linear model it is the b in y = mx + b; in an exponential model it is the a in y = a·b^x. Identifying it first makes writing the model much faster. (Oʻzbekcha: "boshlangʻich qiymat" — chiziqlida b, eksponensialda a boʻladi.)

Practice 1

A gym charges a $40 joining fee plus $25 per month. Linear or exponential? Write the cost after x months.