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College Algebra Modeling

A free College Algebra lesson from the “Modeling and Readiness” unit, with a worked example and practice problems including step-by-step solutions.

Modeling means choosing a function type that matches a situation. Linear models use constant difference, exponential models use constant percent change, and quadratics model many maximum or minimum contexts.

What you'll learn

Why it matters: Forecasting sales, comparing depreciation, fitting height data, and choosing an equation for a context all start with model selection. Students need to notice the structure first: constant difference, constant ratio, turning point, restriction, or inverse growth.

Worked example

Problem. A value starts at 400 and grows by 6% each year. What model family fits?

  1. The starting amount is fixed.
  2. The change is a constant percent.
  3. Constant percent change points to exponential growth.

Answer: exponential growth

Practice problems

1. A plan charges a fixed fee plus dollars per month. Which model fits?

Choices: Linear · Quadratic · Exponential · Circle

Show solution
  1. Warm-up: First identify exactly what the question is asking: A plan charges a fixed fee plus dollars per month. Which model fits?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The amount changes by a constant amount.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Linear

2. A population grows by 3% each year. Which model fits?

Choices: Exponential · Linear · Quadratic · Absolute value

Show solution
  1. Core Practice: First identify exactly what the question is asking: A population grows by 3% each year. Which model fits?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Constant percent change.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Exponential

3. A value starts at 250 and decays by 20%. What multiplier is used?

Show solution
  1. Challenge: First identify exactly what the question is asking: A value starts at 250 and decays by 20%. What multiplier is used?
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. 1 - 0.20 = 0.80.
  4. Check the result by substituting or estimating: the response should match 0.8 and make sense in the original problem.

Answer: 0.8

4. Outputs 10, 15, 20, 25 have a constant difference. Which model fits?

Choices: Linear · Exponential · Quadratic · Logarithmic

Show solution
  1. Linear Models: First identify exactly what the question is asking: Outputs 10, 15, 20, 25 have a constant difference. Which model fits?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Constant difference suggests linear.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Linear

5. Outputs 3, 6, 12, 24 have a constant ratio. Which model fits?

Choices: Exponential · Linear · Quadratic · Absolute value

Show solution
  1. Exponential Models: First identify exactly what the question is asking: Outputs 3, 6, 12, 24 have a constant ratio. Which model fits?
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. Constant ratio suggests exponential.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Exponential

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