Tanisha is graphing the function [tex]$f(x)=25\left(\frac{3}{5}\right)^x$[/tex]. She begins by plotting the point [tex]$(1,15)$[/tex]. Which could be the next point she plots on the graph?
[tex]$(2,9)$[/tex]
[tex]$(2,-10)$[/tex]
[tex]$\left(2,14 \frac{2}{5}\right)$[/tex]
[tex]$(2,5)$[/tex]
Answer (2)
Verify that the point (1, 15) lies on the graph of f ( x ) = 25 ( 5 3 ) x .
Calculate f ( 2 ) by substituting x = 2 into the function f ( x ) = 25 ( 5 3 ) x .
Find that f ( 2 ) = 9 .
Conclude that the next point Tanisha could plot is ( 2 , 9 ) .
Explanation
Verifying the given point First, we need to verify that the point (1, 15) lies on the graph of f ( x ) = 25 ( 5 3 ) x . We substitute x = 1 into the function:
Calculation for x=1 f ( 1 ) = 25 ( 5 3 ) 1 = 25 × 5 3 = 5 × 3 = 15 So, the point (1, 15) does indeed lie on the graph.
Finding the next point Next, we need to find the y-coordinate when x = 2 . We substitute x = 2 into the function:
Calculation for x=2 f ( 2 ) = 25 ( 5 3 ) 2 = 25 × ( 5 3 ) × ( 5 3 ) = 25 × 25 9 = 9
Conclusion Therefore, the point (2, 9) lies on the graph of the function.
Examples
Exponential functions like the one in this problem are used to model various real-world phenomena, such as the decay of radioactive substances, the growth of populations, and the depreciation of assets. For instance, if you invest money in a savings account with a fixed interest rate compounded annually, the growth of your investment can be modeled using an exponential function. Similarly, if you purchase a car, its value depreciates over time, and this depreciation can often be modeled using an exponential function.
The next point Tanisha could plot after (1, 15) is (2, 9), as calculated from the function f ( x ) = 25 ( 5 3 ) x . This was determined by substituting x = 2 into the function. Therefore, the answer is ( 2 , 9 ) .
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