IM Algebra 2 - Unit 4 Mid Review

An exponential function is a mathematical function of the form $$f(x) = a \times b^x$$ , where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It describes growth or decay processes.

The decay factor is the constant ratio by which a quantity decreases in each time period. It is represented as $$b$$ in the function $$f(t) = a \times b^t$$ , where $$0 < b < 1$$ .

The half-life is the time required for a quantity to reduce to half its initial value. It is commonly used in the context of radioactive decay and pharmacokinetics.

The equation for exponential growth is given by $$f(t) = a \times (1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

The initial value in an exponential function is the value of the function when time (or the independent variable) is zero, represented as $$f(0) = a$$ .

If a population triples in a decade, it means that the population at the end of 10 years is three times the population at the start of that decade.

The equation for a half-life problem is structured as $$f(t) = a \times (0.5)^{\frac{t}{T_{1/2}}}$$ , where 'T_{1/2}' is the half-life.

The formula for exponential decay is given by $$f(t) = a \times e^{-kt}$$ , where 'a' is the initial amount, 'k' is the decay constant, and 't' is time.

Doubling time is the period it takes for a quantity to double in size or value. It can be calculated using the rule of 70: $$T_d = \frac{70}{r}$$ , where 'r' is the growth rate.

The base in an exponential function determines the rate of growth or decay. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.