Graph the exponential model $y=2(0.5)^x$. Which point lies on the graph?
(0,2)
(2,5)
(-4,1)
(1,-1)
Answer (1)
Substitute the x -coordinate of each point into the equation y = 2 ( 0.5 ) x .
Calculate the corresponding y value for each x .
Check if the calculated y value matches the y -coordinate of the given point.
The point ( 0 , 2 ) satisfies the equation, so the answer is ( 0 , 2 ) .
Explanation
Understanding the Problem We are given the exponential model y = 2 ( 0.5 ) x and four points: ( 0 , 2 ) , ( 2 , 5 ) , ( − 4 , 1 ) , and ( 1 , − 1 ) . Our goal is to determine which of these points lies on the graph of the given exponential model. A point lies on the graph if substituting its x -coordinate into the equation yields the point's y -coordinate.
Testing Each Point Let's test each point:
Point ( 0 , 2 ) : Substitute x = 0 into the equation: y = 2 ( 0.5 ) 0 = 2 ( 1 ) = 2 . Since the calculated y value is 2, which matches the y -coordinate of the point, the point ( 0 , 2 ) lies on the graph.
Point ( 2 , 5 ) : Substitute x = 2 into the equation: y = 2 ( 0.5 ) 2 = 2 ( 0.25 ) = 0.5 . Since the calculated y value is 0.5, which is not equal to 5, the point ( 2 , 5 ) does not lie on the graph.
Point ( − 4 , 1 ) : Substitute x = − 4 into the equation: y = 2 ( 0.5 ) − 4 = 2 ( 2 4 ) = 2 ( 16 ) = 32 . Since the calculated y value is 32, which is not equal to 1, the point ( − 4 , 1 ) does not lie on the graph.
Point ( 1 , − 1 ) : Substitute x = 1 into the equation: y = 2 ( 0.5 ) 1 = 2 ( 0.5 ) = 1 . Since the calculated y value is 1, which is not equal to -1, the point ( 1 , − 1 ) does not lie on the graph.
Conclusion Therefore, the point ( 0 , 2 ) lies on the graph of the exponential model y = 2 ( 0.5 ) x .
Examples
Exponential models are used to describe phenomena that decay over time, such as the depreciation of a car's value. For example, if a car's initial value is 20 , 000 an d i t d e p rec ia t es a t a r a t eo f 50 x ye a rsc anb e m o d e l e d b y t h ee q u a t i o n y = 20000(0.5)^x$. Understanding exponential decay helps in making informed decisions about investments, asset management, and predicting future trends based on current data.
