The graph represents the function $f(x)=10(2)^x$.
How would the graph change if the $b$ 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 $y$-axis.
B. The graph will increase at a faster rate.
C. The graph will increase at a slower rate.
D. The $y$-values will continue to increase as $x$-increases.
E. The $y-values$ will each be less than their corresponding $x$-values.
Answer (2)
When the base value of the function f ( x ) = 10 ( 2 ) x is decreased but remains greater than 1, the y-intercept stays the same. The graph will increase at a slower rate due to the smaller base, but the y-values will continue to increase as x increases. Therefore, the correct options are that the graph will increase at a slower rate and that the y-values will continue to increase as x increases.
;
The graph will begin at the same point on the y -axis because f ( 0 ) = 10 regardless of the value of b .
The graph will increase at a slower rate because the base b is smaller.
The y -values will continue to increase as x increases because 1"> b > 1 .
The y -values will not necessarily be less than their corresponding x -values.
The correct options are:
The graph will increase at a slower rate. The y -values will continue to increase as x -increases.
Explanation
Problem Analysis The problem describes an exponential function f ( x ) = 10 ( 2 ) x and asks how its graph changes when the base b = 2 is decreased, but remains greater than 1. Let's analyze each option.
Analyzing the y-intercept The first option states: 'The graph will begin at a lower point on the y -axis.' The y -intercept is the point where the graph intersects the y -axis, which occurs when x = 0 . For the original function, f ( 0 ) = 10 ( 2 ) 0 = 10 ( 1 ) = 10 . If we decrease b to some value b ′ such that 1 < b ′ < 2 , then the new function is f ( x ) = 10 ( b ′ ) x . The y -intercept of the new function is f ( 0 ) = 10 ( b ′ ) 0 = 10 ( 1 ) = 10 . Since the y -intercept remains the same, this option is incorrect.
Analyzing the rate of increase The second option states: 'The graph will increase at a faster rate.' The rate of increase of an exponential function is determined by the base b . Since we are decreasing b , the rate of increase will be slower, not faster. Therefore, this option is incorrect.
Confirming the slower rate of increase The third option states: 'The graph will increase at a slower rate.' As explained in the previous step, decreasing the base b will indeed cause the graph to increase at a slower rate. Therefore, this option is correct.
Analyzing the increasing y-values The fourth option states: 'The y -values will continue to increase as x -increases.' Since the new base b ′ is still greater than 1 ( 1 < b ′ < 2 ), the function f ( x ) = 10 ( b ′ ) x will still be an increasing function. As x increases, the y -values will also increase. Therefore, this option is correct.
Analyzing the relationship between y and x values The fifth option states: 'The y − v a l u es will each be less than their corresponding x -values.' This statement is not generally true for exponential functions. For example, if x = 1 and b ′ = 1.5 , then f ( 1 ) = 10 ( 1.5 ) 1 = 15 , which is greater than x = 1 . Therefore, this option is incorrect.
Final Answer Therefore, the correct options are:
The graph will increase at a slower rate.
The y -values will continue to increase as x -increases.
Examples
Exponential functions are used to model population growth. If a population's growth rate decreases (corresponding to a decrease in the base 'b' of the exponential function), the population will still grow, but at a slower pace. This concept is crucial in understanding demographic changes and resource management. For example, if a city's population is modeled by P ( t ) = 1000 ( 1.1 ) t , decreasing the growth rate to 1.05 would change the model to P ( t ) = 1000 ( 1.05 ) t , resulting in slower population growth over time.
