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Learning Goal
Part of: Distinguish linear and exponential functions — 1 of 3 group items
Prove growth properties of functions
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.
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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
Prerequisites
Slides
Interactive presentations perfect for visual learners • 2 slide decks
Slide Video
Watch narrated slides play like a video lesson • Narrated slide playback
Task-sets
Learning resource • 1 task-sets