Pherris is graphing the function [tex]$f(x)=2(3)^x$[/tex]. He begins with the point [tex]$(1,6)$[/tex]. Which could be the next point on his graph?
[tex]$(2,12)$[/tex]
[tex]$(2,18)$[/tex]
[tex]$(2,7)$[/tex]
[tex]$(3,7)$[/tex]
Answer (1)
The problem asks us to find a point that lies on the graph of the function f ( x ) = 2 ( 3 ) x .
We test each point by substituting its x-coordinate into the function and checking if the result matches the y-coordinate.
For the point (2, 18), we have f ( 2 ) = 2 ( 3 ) 2 = 18 , which matches the y-coordinate.
Therefore, the point (2, 18) lies on the graph of the function. ( 2 , 18 )
Explanation
Understanding the Problem We are given the function f ( x ) = 2 ( 3 ) x and the point ( 1 , 6 ) which lies on the graph. We need to determine which of the given points could be the next point on the graph. The given options are ( 2 , 12 ) , ( 2 , 18 ) , ( 2 , 7 ) , and ( 3 , 7 ) .
Checking the Points To check if a point ( x , y ) lies on the graph of the function f ( x ) = 2 ( 3 ) x , we need to substitute the x -coordinate into the function and see if the result matches the y -coordinate.
Checking (2, 12) Let's check the point ( 2 , 12 ) . We calculate f ( 2 ) = 2 ( 3 ) 2 = 2 ( 9 ) = 18 . Since 18 e q 12 , the point ( 2 , 12 ) is not on the graph.
Checking (2, 18) Now let's check the point ( 2 , 18 ) . We calculate f ( 2 ) = 2 ( 3 ) 2 = 2 ( 9 ) = 18 . Since 18 = 18 , the point ( 2 , 18 ) is on the graph.
Checking (2, 7) Next, let's check the point ( 2 , 7 ) . We calculate f ( 2 ) = 2 ( 3 ) 2 = 2 ( 9 ) = 18 . Since 18 e q 7 , the point ( 2 , 7 ) is not on the graph.
Checking (3, 7) Finally, let's check the point ( 3 , 7 ) . We calculate f ( 3 ) = 2 ( 3 ) 3 = 2 ( 27 ) = 54 . Since 54 e q 7 , the point ( 3 , 7 ) is not on the graph.
Conclusion Therefore, the only point among the given options that lies on the graph of the function f ( x ) = 2 ( 3 ) x is ( 2 , 18 ) .
Examples
Exponential functions like f ( x ) = 2 ( 3 ) x are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For example, if a bacterial population triples every hour and you start with 2 bacteria, the function f ( x ) = 2 ( 3 ) x models the population after x hours. Finding points on the graph helps predict the population size at different times. Understanding exponential growth is crucial in fields like biology, finance, and environmental science.
