Recognizing linear pattern
Constant difference indicates linear behavior.
Recognizing exponential pattern
Constant ratio indicates exponential behavior.
Inverse function — definition?
Reverses input-output relationship of original function.
Finding inverse of linear
Solve for x, then swap x and y.
Inverse of exponential — method?
Use logarithms to solve for x.
Graph of inverse — reflection?
Reflection across y = x line.
Percent change — calculation?
(Final - Initial) / Initial × 100%.
Multi-year growth to per-year?
Take root of multi-year growth factor.
Exponential model form?
f(t) = A · b^t.
Solve exponential with logs
Isolate exponential, then apply logarithm.
Logarithm product rule
log_b(xy) = log_b x + log_b y.
Logarithm quotient rule
log_b(x/y) = log_b x - log_b y.
Logarithm power rule
log_b(x^k) = k log_b x.
Transformations — horizontal shift
Replace x with x - h in function.
Transformations — vertical shift
Add or subtract constant to function.
Transformations — reflection
Multiply function by -1.
Transformations — stretch/compression
Multiply function by a > 1 (stretch) or 0 < a < 1 (compress).
Complex number form
a + bi, with real a, b.
Addition of complex numbers
Add real parts and imaginary parts separately.
Multiplication of complex numbers
Use distributive property, simplify with i^2 = -1.
Factoring difference of squares
a^2 - b^2 = (a - b)(a + b).
Quadratic with complex roots
Form from conjugates: (x - (a+bi))(x - (a-bi)).
Teste seu conhecimento com 11 perguntas sobre Understanding Exponential and Linear Functions.
1. What is the primary purpose of linear and exponential functions in data modeling?
2. A company models its population growth with the function P(t) = 500 * 1.2^t, where t is in years. If the population after some years is 1,000, how can you find the number of years t?
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