The graph represents the function [tex]$f(x)=10(2)^x$[/tex].
How would the graph change if the [tex]$b$[/tex] value in the equation is decreased but remains greater than 1? Check all that apply.
A. The graph will begin at a lower point on the [tex]$y$[/tex]-axis.
B. The graph will increase at a faster rate.
C. The graph will increase at a slower rate.
D. The [tex]$y$[/tex]-values will continue to increase as [tex]$x$[/tex]-increases.
E. The [tex]$y$[/tex]-values will each be less than their corresponding [tex]$x$[/tex]-values.
Answer (2)
The y-intercept remains unchanged.
The graph increases at a slower rate.
The y-values continue to increase as x increases.
The y-values are less than their corresponding values of the original function. The graph will increase at a slower rate, The y-values will continue to increase as x-increases, The y-values will each be less than their corresponding values of the original function.
Explanation
Understanding the Problem The problem describes an exponential function f ( x ) = 10 ( 2 ) x and asks how the graph changes if the base, 2, is decreased to a value b ′ such that 1 < b ′ < 2 . We need to analyze how this change affects the graph's characteristics.
Analyzing the y-intercept The y -intercept occurs when x = 0 . For the original function, f ( 0 ) = 10 ( 2 ) 0 = 10 ( 1 ) = 10 . For the modified function, f ( x ) = 10 ( b ′ ) x , the y -intercept is f ( 0 ) = 10 ( b ′ ) 0 = 10 ( 1 ) = 10 . Thus, the y -intercept remains unchanged.
Comparing y-values Since 1 < b ′ < 2 , for any 0"> x > 0 , we have ( b ′ ) x < 2 x . Therefore, 10 ( b ′ ) x < 10 ( 2 ) x . This means that for any positive x , the y -value of the modified function is less than the y -value of the original function.
Analyzing the rate of increase Since 1"> b ′ > 1 , the function f ( x ) = 10 ( b ′ ) x is still an increasing function. As x increases, y also increases. However, because b ′ < 2 , the rate of increase is slower than the original function.
Conclusion Based on the analysis, the correct statements are:
The graph will increase at a slower rate.
The y -values will continue to increase as x -increases.
The y -values will each be less than their corresponding values of the original function.
Examples
Consider a scenario where you are modeling population growth. The original model predicts a doubling of the population every year, represented by the function f ( x ) = 10 ( 2 ) x , where x is the number of years and f ( x ) is the population size (in millions). If, due to resource constraints, the growth rate slows down such that the population increases by a factor less than 2 (but still more than 1) each year, the new growth can be modeled by f ( x ) = 10 ( 1.5 ) x . This means the population still grows, but at a slower rate, and the population size at any given year will be less than what the original model predicted. This kind of modeling is crucial for resource planning and policy making.
When the base value in the function f ( x ) = 10 ( 2 ) x is decreased while remaining greater than 1, the graph increases at a slower rate, the y-values continue to rise, and they will be lower than those of the original function for the same x-values. This indicates that the growth is less rapid than before. The correct choices are C, D, and E.
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