Pherris is graphing the function [tex]$f(x)=2(3)^x$[/tex]. He begins with the point (1,6). Which could be the next point on his graph?
A. (2,12)
B. (2,18)
C. (2,7)
D. (3,7)
Answer (2)
The problem gives the function f ( x ) = 2 ( 3 ) x and asks to find the next point on the graph, given the point ( 1 , 6 ) .
Check if the point ( 2 , 12 ) satisfies the equation: f ( 2 ) = 2 ( 3 ) 2 = 18 = 12 .
Check if the point ( 2 , 18 ) satisfies the equation: f ( 2 ) = 2 ( 3 ) 2 = 18 .
Check if the point ( 2 , 7 ) satisfies the equation: f ( 2 ) = 2 ( 3 ) 2 = 18 = 7 .
Check if the point ( 3 , 7 ) satisfies the equation: f ( 3 ) = 2 ( 3 ) 3 = 54 = 7 .
The point ( 2 , 18 ) is the only point that satisfies the equation, so the answer is ( 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 of the function. We need to determine which of the given points could be the next point on the graph. To do this, we will check each point to see if it satisfies the equation.
Checking (2,12) Let's check the point ( 2 , 12 ) . We need to see if f ( 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 need to see if f ( 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 need to see if f ( 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 need to see if f ( 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 The only point that satisfies the equation f ( x ) = 2 ( 3 ) x is ( 2 , 18 ) . Therefore, the next point on the graph could be ( 2 , 18 ) .
Examples
Exponential functions like the one in this problem are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have will grow exponentially over time. Understanding how to graph and analyze exponential functions can help you make informed decisions about your investments.
The next point on the graph of the function f ( x ) = 2 ( 3 ) x after the point ( 1 , 6 ) is ( 2 , 18 ) . This is verified by calculating f ( 2 ) = 18 . The correct option is therefore ( 2 , 18 ) .
;
