Final answer:
To identify whether an equation represents exponential growth or decay, look at the base of the exponent; bases bigger than 1 indicate growth, and bases between 0 and 1 indicate decay. Exponential growth is seen in population increases or financial returns, while exponential decay can be observed in radioactive substances or asset depreciation.
Step-by-step explanation:
In mathematics, particularly in algebra, we often deal with equations that represent either exponential growth or exponential decay. To discern which type an equation represents, we examine the base of the exponent. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1 (but not 0 or 1 itself), it represents exponential decay.
- a. w= 9/8(3/5)ᵗ - Since 3/5 is less than 1, this equation represents exponential decay.
- b. L=0.25(12)ᵗ - 12 is greater than 1, indicating this equation is exponential growth.
- c. f(t) = (3/4)ᵗ - 3/4 is less than 1, showing exponential decay.
- d. w=0.5 (2.1)ᵗ - Here, 2.1 is greater than 1, which means this is exponential growth.
- e. L=4.2(0.6)ᵗ - Since 0.6 is less than 1, e represents exponential decay.
- f. f(t) = 2/3(6)ᵗ - The base 6 exceeds 1, so this function depicts exponential growth.
- g. g(x)=1.3(x) - This final equation is not exponential at all, because the exponent is missing; instead, it represents linear growth.
Exponential growth occurs when a population or quantity increases at a rate proportional to its current value, such as in populations where the birth rate leads to more individuals to reproduce over time, or in financial investments with a consistent return rate. Conversely, exponential decay occurs when a quantity decreases at a rate proportional to its current value, evident in radioactive decay or depreciation of assets over time.