Comparing Linear, Exponential, and Quadratic

Learn and apply comparing linear, exponential, and quadratic using equations, examples, and interpretation.

Lesson 5 of 53 parts Practice Flashcards

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First Differences

Objective: Explain and use first differences in an Algebra 1 problem.

Equal first differences over equal input steps indicate a linear pattern in a table. No, many functions can match finitely many rows.

Worked examples

Try itExample 1.1: In a linear table, outputs at x=0 and x=1 are 13 and 15. What is the first difference?Show answer

2. Subtract the first output from the second. The answer is 2.

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First Differences

Objective: Explain and use first differences in an Algebra 1 problem.

Equal first differences over equal input steps indicate a linear pattern in a table. No, many functions can match finitely many rows.

Worked examples

Try itExample 1.1: In a linear table, outputs at x=0 and x=1 are 13 and 15. What is the first difference?Show answer

2. Subtract the first output from the second. The answer is 2.

Second Differences

Objective: Explain and use second differences in an Algebra 1 problem.

Equal second differences over equal input steps suggest a quadratic rule for the table under a polynomial model. The simple difference test assumes equal steps.

Worked examples

Try itExample 2.1: For y=(x−8)²+23, what is the vertex x-coordinate?Show answer

8. Vertex form reveals the horizontal coordinate where the square is zero. The answer is 8.

Ratios and Context

Objective: Explain and use ratios and context in an Algebra 1 problem.

Equal nonzero output ratios over equal steps suggest an exponential rule; choose among models using pattern and meaning. A numerical pattern may be unreasonable outside its context.

Worked examples

Try itExample 3.1: A quantity starts at 33 and increases by factor 4 per period. What is its value after 11 periods?Show answer

138412032. Multiply the initial amount by the factor once per period. The answer is 138412032.

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