Flashcard: Understanding Exponential and Linear Functions — 22 carte

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)).