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Modeling with Functions

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

Modeling means choosing a function family that matches a situation. Linear models have constant change, quadratic models have turning points, and exponential models have constant percent change.

What you'll learn

Why it matters: Forecasting sales, fitting a temperature curve, and predicting depreciation all start with picking a function family. Linear, quadratic, exponential, or sinusoidal — match the shape of the data first, then fit the parameters.

Worked example

Problem. A population starts at 500 and grows by 8% each year. Which model fits?

  1. The starting amount is 500.
  2. Growth by 8% is constant percent change.
  3. Constant percent change points to an exponential model.

Answer: exponential growth

Practice problems

1. A taxi fare increases by the same dollar amount per mile. Which model fits?

Choices: Linear · Quadratic · Exponential · Trigonometric

Show solution
  1. Warm-up: First identify exactly what the question is asking: A taxi fare increases by the same dollar amount per mile. Which model fits?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Constant rate of change.
  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 ball's height rises and then falls. Which model often fits?

Choices: Quadratic · Linear · Exponential · Rational only

Show solution
  1. Warm-up: First identify exactly what the question is asking: A ball's height rises and then falls. Which model often fits?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Projectile height is often quadratic.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Quadratic

3. A bank account grows by 4% 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 bank account grows by 4% 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

4. A value starts at 120 and decays by 15%. What multiplier is used?

Show solution
  1. Challenge: First identify exactly what the question is asking: A value starts at 120 and decays by 15%. What multiplier is used?
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. Decay multiplier is 1 - 0.15.
  4. Check the result by substituting or estimating: the response should match 0.85 and make sense in the original problem.

Answer: 0.85

5. Outputs 3, 6, 12, 24 suggest which model family?

Choices: Exponential · Linear · Quadratic · Constant

Show solution
  1. Mixed Review: First identify exactly what the question is asking: Outputs 3, 6, 12, 24 suggest which model family?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Each output is multiplied by 2.
  4. A constant ratio points to an exponential model.
  5. So the model family is exponential.
  6. 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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