Prove growth properties of functions

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Lesson Plan →Objectives, pacing and practice, built from this lesson's brief.
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HSF.LE.A.1.a

**HSF.LE.A.1.a**: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

HSF.LE.A.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

What you'll learn

Prove algebraically that for f(x) = mx + b, the difference f(x + d) - f(x) = md for any equal spacing d, showing the result is independent of x
Prove algebraically that for g(x) = a * b^x, the ratio g(x + d) / g(x) = b^d for any equal spacing d, showing the result is independent of x
Explain why the constant difference property is equivalent to a constant rate of change (slope)
Explain why the constant ratio property is equivalent to a constant growth/decay factor (base)
Use the proofs to determine whether a given function is linear or exponential without relying solely on tables of values
Connect the algebraic proofs to numerical examples by verifying the formulas with specific values of x and d

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