given the following exponential function, identify whether the change represents growth or decay, and…

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease. \n( y = 51(0.91)^x )

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease. \n( y = 51(0.91)^x )

Answer

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is ( y = a(b)^x ), where ( a ) is the initial amount and ( b ) is the base. If ( b > 1 ), it's growth; if ( 0 < b < 1 ), it's decay. Here, ( b = 0.91 ), which is between 0 and 1, so it's decay.

Step2: Calculate percentage rate of decrease

The formula for the rate of change is ( \text{Rate} = (1 - b) \times 100% ) for decay (since ( b = 1 - r ), where ( r ) is the rate of decrease). Substitute ( b = 0.91 ): ( \text{Rate} = (1 - 0.91) \times 100% = 0.09 \times 100% = 9% ) decrease.

Answer:

Decay, 9% decrease