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In the context of exponential functions, the starting value is a crucial concept. It represents the initial quantity or size of something before any growth or decay happens.
The starting value is typically denoted by the letter \( a \) in the exponential function expressed as \( Q(t) = a \cdot b^t \). It's the constant that the function starts with when \( t = 0 \).
In simpler terms, if you have a population of rabbits starting at 100, then the starting value \( a \) would be 100 for the population function. In our problem, the starting value is quite obvious.
Re-examining the equation \( Q = \frac{210}{3 \cdot 2^t} \), which was rearranged to \( Q = 210 \cdot (2^{-t}) \cdot \frac{1}{3} \), allows us to identify the starting value as \( 210 \).

• Given value: 210
• Represents: Initial size or quantity

The starting value establishes the baseline from which all increases or decreases are measured. Understanding this helps in analyzing the trend and interpreting the results.

The growth factor in an exponential function determines how the function increases or decreases over time. It's denoted by \( b \) in the expression \( Q(t) = a \cdot b^t \). This factor tells us by what proportion or factor the starting value changes with each increment in \( t \).
A growth factor greater than 1 indicates growth, while a factor between 0 and 1 indicates decay. Consider a scenario where a quantity doubles every time period; the growth factor would be 2.
For our given function, the task was to identify the growth factor from \( Q = 210 \cdot (2^{-t}) \cdot \frac{1}{3} \). Here, the growth factor \( b = 2^{-1} \cdot \frac{1}{3} \), which simplifies to \( \frac{1}{6} \). This indicates the quantity decreases with time, as our growth factor is less than one.

• Calculated as: \( 2^{-1} \cdot \frac{1}{3} = \frac{1}{6} \)
• Indicates: Decrease by a factor of six over time

The importance of the growth factor lies in its ability to predict future values based on past data, making it key for understanding exponential trends in various fields.

The percentage growth rate is a way to represent the change in a quantity as a percentage, making it easier to understand how rapidly or slowly something grows or decays. It is calculated from the growth factor \( b \), using the formula \( r = (b - 1) \times 100 \).
For our exponential function, the growth factor is \( \frac{1}{6} \). Substituting into the formula gives \( r = (\frac{1}{6} - 1) \times 100 \), which calculates to \( -\frac{500}{6} \) or approximately -83.33\%. A negative percentage growth rate signifies a decrease or decay.

• Formula: \( r = (b - 1) \times 100 \)
• Applied to: \( b = \frac{1}{6} \)
• Result: Approximately -83.33\%

The percentage growth rate provides a clear way to communicate the rate at which a quantity decreases or increases. This makes it a versatile tool, enabling people to easily compare different growth scenarios, whether in populations, investments, or other applications.