Sure, let's analyze how the graph of the function [tex]\( f(x) = 10(2)^x \)[/tex] would change if the base [tex]\( b = 2 \)[/tex] is decreased but remains greater than 1.
1. The graph will begin at a lower point on the [tex]$y$[/tex]-axis:
Reducing the base [tex]\( b \)[/tex] (making it less than 2 but still greater than 1), does not affect the initial point determined by [tex]\( f(0) = 10 \cdot (2)^0 = 10 \)[/tex]. The value at [tex]\( x = 0 \)[/tex] remains the same for any exponential function [tex]\( f(x) = 10(b)^x \)[/tex] with the same coefficient and varying base [tex]\( b \)[/tex]. So this statement is false.
2. The graph will increase at a faster rate:
If the base [tex]\( b \)[/tex] is decreased but still greater than 1, the rate at which the graph increases will slow down compared to when [tex]\( b = 2 \)[/tex]. Thus, the statement that the graph will increase at a faster rate is false.
3. The graph will increase at a slower rate:
As [tex]\( b \)[/tex] is decreased (but remains greater than 1), the exponential growth slows down. This means the function will increase at a slower rate. So this statement is true.
4. The [tex]$y$[/tex]-values will continue to increase as [tex]$x$[/tex]-increases:
Despite the base [tex]\( b \)[/tex] being decreased but greater than 1, the function [tex]\( f(x) = 10(b)^x \)[/tex] still represents an exponential function. Exponential functions with bases greater than 1 increase as [tex]\( x \)[/tex] increases. Therefore, this statement is true.
5. The [tex]$y$[/tex]-values will each be less than their corresponding [tex]$x$[/tex]-values:
This statement suggests a comparison where for any [tex]\( x \)[/tex], the [tex]\( y \)[/tex]-value (i.e., [tex]\( f(x) \)[/tex]) is less than [tex]\( x \)[/tex]. However, in the case of the exponential function [tex]\( f(x) = 10(b)^x \)[/tex] where [tex]\( b \)[/tex] is greater than 1, there are always points where [tex]\( f(x) \)[/tex] will exceed [tex]\( x \)[/tex], especially when [tex]\( x \)[/tex] becomes sufficiently large. So this statement is false.
In summary:
- Starts at lower y-point: False
- Increases at faster rate: False
- Increases at slower rate: True
- y-values increase as x increases: True
- y-values less than corresponding x-values: False
So, the correct options are:
- The graph will increase at a slower rate.
- The [tex]$y$[/tex]-values will continue to increase as [tex]$x$[/tex]-increases.
