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\n\\( y = 410(0.58)^x \\)

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.\n\n\\( y = 410(0.58)^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 = 0.58 ), and ( 0<0.58<1 ), so it's decay.

Step2: Calculate percentage rate of decrease

The formula for the rate of decrease is ( r=1 - b ) (expressed as a percentage). So ( r = 1 - 0.58 = 0.42 ), which is ( 0.42\times100% = 42% ).

Answer:

The function represents decay with a percentage rate of decrease of ( 42% ).