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)
Substitute x = 2 into the function f ( x ) = 2 ( 3 ) x and calculate f ( 2 ) = 2 ( 3 ) 2 = 18 .
Check if the point ( 2 , 12 ) satisfies the equation: f ( 2 ) = 12 , but 18 = 12 .
Check if the point ( 2 , 18 ) satisfies the equation: f ( 2 ) = 18 , and 18 = 18 .
The point ( 2 , 18 ) is on the graph, 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 ) on the graph. We need to find another point on the graph from the given options: ( 2 , 12 ) , ( 2 , 18 ) , ( 2 , 7 ) , ( 3 , 7 ) .
Checking the Points To determine which of the given points lies on the graph of the function, we need to check if the coordinates of the point satisfy the equation f ( x ) = 2 ( 3 ) x .
Checking (2,12) Let's check the point ( 2 , 12 ) . We need to see if f ( 2 ) = 12 . 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 . f ( 2 ) = 2 ( 3 ) 2 = 2 ( 9 ) = 18 Since 18 = 18 , the point ( 2 , 18 ) is on the graph.
Checking (2,7) Let's check the point ( 2 , 7 ) . We need to see if f ( 2 ) = 7 . 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 . 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 next point on the graph 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 culture starts with 2 bacteria and triples every hour, the function f ( x ) = 2 ( 3 ) x models the number of bacteria after x hours. Understanding how to graph and analyze these functions helps us predict and understand these phenomena.
The next point on the graph of the function f ( x ) = 2 ( 3 ) x after (1, 6) is (2, 18), as substituting x = 2 into the function yields f ( 2 ) = 18 . The correct answer among the options provided is (2, 18).
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