The table represents an exponential function.
x –3 –2 –1 0 1 2 3
y 1 over 1000 1 over 100 1 over 10 1 10 100 1,000
Does the function in the table represent growth or decay?
The function represents exponential decay because the base equals 10.
The function represents exponential growth because the base equals 10.
The function represents exponential decay because the base equals 1 over 10.
The function represents exponential growth because the base equals 1 over 10
The Correct Answer and Explanation is :
The function represented in the table indicates exponential growth because the base is ( \frac{1}{10} ), which is less than 1. To better understand this, let’s analyze the table:
- Values of x and y:
- When ( x = -3 ), ( y = \frac{1}{1000} )
- When ( x = -2 ), ( y = \frac{1}{100} )
- When ( x = -1 ), ( y = \frac{1}{10} )
- When ( x = 0 ), ( y = 1 )
- When ( x = 1 ), ( y = 10 )
- When ( x = 2 ), ( y = 100 )
- When ( x = 3 ), ( y = 1000 )
As the value of ( x ) increases, the value of ( y ) also increases significantly, suggesting that the function is rising. In an exponential function, if the base is greater than 1, it represents growth, while if the base is between 0 and 1, it typically represents decay.
In this case, the values of ( y ) are derived from the base ( \frac{1}{10} ). The function can be expressed as ( y = \left( \frac{1}{10} \right)^{-x} ). As ( x ) takes on negative values, the function outputs large positive values for ( y ) (i.e., ( 1000, 100, 10 )), indicating growth. Conversely, for positive ( x ), the function outputs progressively larger ( y ) values (i.e., ( 10, 100, 1000 )), reinforcing the growth pattern.
In summary, even though the base ( \frac{1}{10} ) might imply decay when considered alone, in this specific context of increasing negative ( x ) values leading to increasing ( y ), it results in exponential growth. Thus, the correct interpretation of the function is that it represents exponential growth because the values of ( y ) grow larger as ( x ) increases.