Question

exponential equations
what is an exponential equation?
exponential equations are equations written in the format,
$y = ab^x$
y is the output value
a is the principal
b is the growth or decay rate
x is the time
define: (use the internet to define and research the following vocabulary)
principal -
____________________
growth –
____________________
decay –
____________________

words associated with growth	words associated with decay
_____________________________	____________________________

Explanation:

• Principal: In the context of exponential equations (and often in finance/growth models), the principal is the initial amount or starting value. For example, in compound interest, it's the initial sum of money invested or borrowed. In \( y = ab^x \), \( a \) (the principal) is the value of \( y \) when \( x = 0 \) (since \( b^0 = 1 \), so \( y = a\times1=a \)).
• Growth: In exponential terms, growth occurs when the base \( b \) in \( y = ab^x \) is greater than 1 (\( b>1 \)). This means that as time (\( x \)) increases, the output value \( y \) increases. For example, population growth, where the number of individuals increases over time, or compound interest where money grows over time.
• Decay: Exponential decay occurs when the base \( b \) in \( y = ab^x \) is between 0 and 1 (\( 0 < b < 1 \)). As time (\( x \)) increases, the output value \( y \) decreases. Examples include radioactive decay (where the amount of a radioactive substance decreases over time) or depreciation of an asset (where the value of a car or machine decreases over time).

Answer:

• Principal: The initial amount or starting value in an exponential (or financial/growth) model; in \( y = ab^x \), it is the value of \( y \) when \( x = 0 \) (i.e., \( a \)).
• Growth: A process where a quantity (represented by \( y \) in \( y = ab^x \)) increases over time (\( x \)) when the growth factor \( b>1 \) (e.g., population growth, compound interest).
• Decay: A process where a quantity (represented by \( y \) in \( y = ab^x \)) decreases over time (\( x \)) when the decay factor \( 0 < b < 1 \) (e.g., radioactive decay, asset depreciation).