Math Models (Modeling but without Integrating Essential Skills) questions test your ability to:
- Translate situations into mathematical language (equations, graphs, tables)
- Build models (functions, expressions) from patterns or scenarios
- Interpret parts of models meaningfully (like the slope in a linear model) These are often abstract or semi-contextual problems, not full-on real-world scenarios.
Creating and Using Equations to Model Relationships
Key Concepts
- Representing relationships with equations or functions
- Choosing the correct equation to fit a pattern or table
- Understanding how different function types behave (linear, quadratic, exponential)
Skills Tested
- Recognizing structure in equations
- Matching equations to described behaviors
- Constructing a function from values or a rule
Example Question
Which of the following functions best models a constant rate of increase? A. f(x) = x^2 + 3 B. f(x) = 2^x C. Answer: f(x) = 3x + 1 (Linear growth = constant rate)
Interpreting Parts of Mathematical Models
Key Concepts
- Understanding the meaning of coefficients, variables, and constants in context
- Interpreting slope, intercepts, and rates of change
Skills Tested
- Identifying what each part of a function represents
- Explaining real-world significance (e.g., “The 0.5 represents the cost per item”)
- Analyzing units and what they imply
Example Question
In the equation C = 25x + 100, where C is total cost and x is the number of items, what does 100 represent? A. The cost per item B. Answer: The fixed initial cost (100 is the constant term: a starting value)
Matching Graphs to Equations
Key Concepts
- Identifying key features of functions: intercepts, asymptotes, shape
- Linear vs. exponential vs. quadratic curves
- Graph transformations: shifts, stretches, reflections
Skills Tested
- Selecting the correct graph for a given equation (or vice versa)
- Understanding how equations change graph shape
- Recognizing key points like vertex, slope, asymptote
Example Question
Which of the following graphs represents the function y = -x^2 + 4? A. Opens upward with vertex at (0, 4) B. Answer: Opens downward with vertex at (0, 4) (Negative leading coefficient = downward parabola)
Using Functions or Expressions to Represent Quantities
Key Concepts
- Writing an expression or equation to model a defined relationship
- Using variables to define unknowns
- Writing equations from sequences or descriptions
Skills Tested
- Constructing expressions from verbal or visual patterns
- Working with function rules or generating equations from a table
Example Question
A sequence begins 3, 7, 11, 15... Which function models the nth term? A. f(n) = 4n + 3 B. Answer: f(n) = 4n - 1 (Common difference is 4 → arithmetic sequence with first term 3)
Comparing and Choosing Appropriate Models
Key Concepts
- Understanding when to use linear vs. quadratic vs. exponential models
- Judging which model best fits a table or graph of data
- Using growth behavior (constant change vs. percentage change) as clues
Skills Tested
- Interpreting trends in data
- Selecting the appropriate model form (e.g., exponential for rapid growth)
- Matching model structure to scenario
Example Question
A population doubles every year. Which model best represents the population P after t years? A. P = 2t B. Answer: P = P_0 \cdot 2^t (Exponential growth: doubling = base 2 exponent)