Growth or Decay - Algebra-1

1. Fundamental Concepts

Nature of exponential growth: After each unit of time, the quantity grows by a factor of b (). For example, natural phenomena such as population growth and bacterial reproduction often conform to this model. The growth factor b reflects the speed of growth (the larger b is, the faster the growth).
Nature of exponential decay: After each unit of time, the quantity decays to a proportion of b () of its original value. For example, the decay of radioactive substances and the depreciation of items. The closer the decay factor b is to 0, the faster the decay rate.
Common characteristics: The function graph is a curve. For growth, it shows an "upward convex" rising trend; for decay, it shows a "downward convex" falling trend. Neither intersects the coordinate axes (since , when ).

Determine whether the following functions represent exponential growth or decay:

A type of bacteria has an initial count of 100, and its quantity doubles every hour (i.e., the growth factor ).

(1) Write the function expression for the number of bacteria y after t hours;

(2) Calculate the number of bacteria after 5 hours.

Solution:

(1) The initial value and the growth factor . The function expression is: .

(2) When , (individuals).

A radioactive substance has an initial mass of 800 grams and decays to half of its original mass each year (i.e., the decay factor ).

(1) Write the function expression for the mass y of the substance after t years;

(2) After how many years will the mass decay to 12.5 grams?

Solution:

(1) The initial value and the decay factor . The function expression is: .

(2) Let , then . Simplifying gives . Since , we have (years).

Model identification:
- If the problem describes a scenario where "the quantity grows by a fixed multiple over time" (e.g., "becomes 1.2 times the original each year"), it corresponds to exponential growth, and b is that multiple ().
- If it describes "the quantity decays by a fixed proportion over time" (e.g., "remains 0.9 times the original each month"), it corresponds to exponential decay, and b is that proportion ().
Parameter extraction:
- Clearly identify the initial value a (usually the quantity when ) and the base b (the fixed growth or decay factor). Avoid confusing "growth/decay ratio" with "factor" (e.g., "grows by 20% each year" corresponds to ; "decays by 30% each month" corresponds to ).
Equation solving:
- When finding the time required for the quantity to reach a certain value, solve the equation using exponential operations or logarithms (if involved). You can first simplify both sides of the equation to the form of powers with the same base, then calculate based on "if the powers are equal and the bases are equal, the exponents are equal" (as in the difficult example).
Rationality verification:
- In a growth model, the result should be greater than the initial value as x increases; in a decay model, the result should be less than the initial value as x increases. This rule can be used to quickly check for calculation errors.