Modeling with Functions Flashcards

Grade 9 · Math · Flashcards

Core strategies, formulas, and examples for building and interpreting mathematical models from real-world data.

Flashcards

Three main function model types

Linear (constant rate of change), Quadratic (parabolic shape, area, projectiles), Exponential (percent growth/decay)

How to spot a linear model

The rate of change (slope) is constant; adding the same amount each period

How to spot an exponential model

Multiplying by the same factor each period; constant percent change

How to spot a quadratic model

The second differences are constant; involves area, squared quantities, or parabolic data

Slope formula from two points

m = (y₂ − y₁) / (x₂ − x₁)

Build linear model from context

Identify the starting value (b) and the rate of change (m), then write f(x) = mx + b

Build exponential model from context

Identify initial value a and growth/decay factor b, then write f(t) = a · bᵗ

Interpolation vs Extrapolation

Interpolation: estimate within the data range. Extrapolation: estimate outside the data range (less reliable)

Domain restriction example

C(n) = 5n + 10 for items n: domain is n ≥ 0 (cannot buy negative items)

Profit model

P(x) = Revenue − Cost; often quadratic or linear depending on context

Percent growth factor

r% annual growth → multiply by (1 + r/100) each period. 6% → factor is 1.06

Percent decay factor

r% annual decay → multiply by (1 − r/100) each period. 10% → factor is 0.90

Finding the model equation from a table

Step 1: check for linear/exponential/quadratic pattern. Step 2: identify parameters. Step 3: write and verify the function.

What does the vertex represent in a quadratic model?

The maximum or minimum output — e.g., the highest point of a projectile or minimum cost

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