Modeling with Functions

Grade 9 · Math · Study Guide

Build and interpret linear, quadratic, and exponential function models from real-world data and context.

Mathematical modeling uses functions to represent real-world situations. The first step is identifying the function type that best fits the data or context. Linear functions (f(x) = mx + b) model situations with a constant rate of change. Quadratic functions (f(x) = ax² + bx + c) model situations involving area, projectile motion, or symmetric patterns. Exponential functions (f(x) = a · bˣ) model growth or decay with a constant percentage change.

To build a linear model, find the rate of change (slope) and the starting value (y-intercept) from the context or data. For two data points, slope m = (y₂ − y₁)/(x₂ − x₁).

Always interpret the parameters in context. In a model like C(t) = 25 + 12t for a cell phone plan, the 25 represents a $25 flat fee and 12 represents $12 per month. Evaluating C(6) = 25 + 72 = $97 tells you the total cost after 6 months.

Domain restrictions often apply in real-world models. A population cannot be negative, time cannot go backward, and some physical quantities have natural upper limits. Always consider whether the mathematical domain of your function makes sense in context before making predictions.

Key terms

Mathematical model
A function or equation used to represent and analyze a real-world situation
Rate of change
How much one quantity changes relative to another; in linear models, this is the slope m
Domain restriction
A limitation on input values based on the real-world context of a model
Linear model
A function f(x) = mx + b used when the rate of change is constant
Quadratic model
A function f(x) = ax² + bx + c used for situations involving a parabolic pattern such as projectile height
Interpreting parameters
Explaining what the coefficients and constants in a function mean in the context of the real-world problem

Flashcards

Identify linear vs exponential from table
Linear: constant first differences. Exponential: constant ratio between consecutive terms
Linear model setup
Find slope m = Δy/Δx, then use y = mx + b with a known point to find b
Interpret slope in context
Slope m means 'for each unit increase in x, y changes by m units'
Interpret y-intercept in context
The y-intercept b means 'the value of y when x = 0 (the starting value)'
Projectile height model
h(t) = −16t² + v₀t + h₀ where v₀ = initial velocity (ft/s) and h₀ = initial height (ft)
Growth rate from exponential model
In f(t) = a(1 + r)ᵗ, the growth rate is r. E.g., b = 1.08 means r = 8%
Evaluate C(m) = 1.50m + 3 at m = 8
C(8) = 1.50(8) + 3 = 12 + 3 = $15
When to use quadratic model
When the data has a parabolic shape, involves area, or describes projectile/freefall motion

Questions

  1. A taxi charges $3.00 flat fee plus $1.50 per mile. Which function models the cost C for m miles?

    1. C(m) = 1.50m
    2. C(m) = 3.00m + 1.50
    3. C(m) = 1.50m + 3.00
    4. C(m) = 4.50m
    Show answer

    Answer: C(m) = 1.50m + 3.00

    The flat fee is the y-intercept ($3.00) and the per-mile charge is the slope ($1.50/mile). C(m) = 1.50m + 3.00.

  2. The height (in feet) of a ball thrown upward is h(t) = −16t² + 48t + 5. The initial height when t = 0 is __ feet.

    Show answer

    Answer: 5

    h(0) = −16(0)² + 48(0) + 5 = 5 feet. The constant term gives the initial height.

  3. A savings account starts with $500 and earns 4% interest annually. Which model represents the balance B after t years?

    1. B(t) = 500 + 0.04t
    2. B(t) = 500(1.04)ᵗ
    3. B(t) = 500(0.04)ᵗ
    4. B(t) = 500 · 4ᵗ
    Show answer

    Answer: B(t) = 500(1.04)ᵗ

    Compound interest is an exponential model. Each year the balance is multiplied by (1 + 0.04) = 1.04, so B(t) = 500(1.04)ᵗ.

  4. In the model P(t) = 200(1.05)ᵗ for a population after t years, the value 1.05 means the population grows by 5% each year.

    Show answer

    Answer: True

    In the model P(t) = 200(1.05)ᵗ, the base is 1.05 = 1 + 0.05, confirming a 5% growth rate per year.

  5. A function models the area A of a square with side length s: A(s) = s². What function type is this?

    1. Linear
    2. Exponential
    3. Quadratic
    4. Absolute value
    Show answer

    Answer: Quadratic

    A(s) = s² is a quadratic function because the variable s is raised to the second power.

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