Exponential Functions Flashcards

Grade 9 · Math · Flashcards

Essential definitions, properties, and examples for mastering exponential functions and their applications.

Flashcards

Standard form of exponential function: f(x) = a · bˣ, where a = initial value, b = base (b > 0, b ≠ 1), x = exponent
Growth factor b > 1: Function grows as x increases. b = 1 + r where r is the growth rate (e.g., b = 1.06 means 6% growth)
Decay factor 0 < b < 1: Function decays as x increases. b = 1 − r where r is the decay rate (e.g., b = 0.85 means 15% decay)
Key point on any exponential graph: (0, a) — when x = 0, f(0) = a · b⁰ = a · 1 = a
Horizontal asymptote of f(x) = a · bˣ: y = 0 (the x-axis), assuming no vertical shift
Horizontal asymptote of f(x) = a · bˣ + k: y = k (the asymptote shifts by the vertical translation k)
Compound interest formula: A = P(1 + r/n)^(nt); P=principal, r=annual rate, n=compoundings/year, t=years
Rule of 72: Approximate years to double = 72 ÷ interest rate%. E.g., at 6%, money doubles in ~12 years
Evaluate f(x) = 2 · 3ˣ at x = 4: f(4) = 2 · 3⁴ = 2 · 81 = 162
Solve 5ˣ = 125: 125 = 5³, so x = 3
Natural base e: Irrational constant ≈ 2.718; used in continuous growth model A = Peʳᵗ
Domain and range of exponential functions: Domain: all real numbers (−∞, ∞); Range: (0, ∞) for basic form, (k, ∞) if shifted up by k
Reflection: f(x) = −bˣ: Reflects across the x-axis; range becomes all negative reals (−∞, 0)