Which properties are present in a table that represents an exponential function in the form [tex]$y=b^x$[/tex] when [tex]$b\ \textgreater \ 1$[/tex]?
I. As the [tex]$x$[/tex]-values increase, the [tex]$y$[/tex] values increase.
II. The point [tex]$(1,0)$[/tex] exists in the table.
III. As the [tex]$x$[/tex]-values increase, the [tex]$y$[/tex]-values decrease.
IV. As the [tex]$x$[/tex]-values decrease, the [tex]$y$[/tex]-values decrease, approaching a singular value.
A. I and IV
B. I and II
C. II and III
D. III only
Answer (1)
Exponential functions y = b x with 1"> b > 1 exhibit specific properties.
As x increases, y increases, indicating exponential growth.
The function does not pass through ( 1 , 0 ) since y = b when x = 1 .
As x decreases, y decreases, approaching 0.
Therefore, the correct answer is I and IV, representing exponential growth and asymptotic behavior towards 0. I an d I V
Explanation
Analyzing the Problem We are given an exponential function y = b x where 1"> b > 1 . We need to determine which of the given properties hold true for this function. Let's analyze each statement.
Evaluating Statement I Statement I: As the x -values increase, the y -values increase. Since 1"> b > 1 , the exponential function is increasing. This means as x increases, y also increases. So, statement I is true.
Evaluating Statement II Statement II: The point ( 1 , 0 ) exists in the table. If x = 1 , then y = b 1 = b . Since 1"> b > 1 , y cannot be 0. Therefore, the point ( 1 , 0 ) does not exist in the table, and statement II is false.
Evaluating Statement III Statement III: As the x -values increase, the y -values decrease. Since 1"> b > 1 , the exponential function is increasing, meaning as x increases, y increases, not decreases. So, statement III is false.
Evaluating Statement IV Statement IV: As the x -values decrease, the y -values decrease, approaching a singular value. As x decreases (i.e., approaches − ∞ ), y = b x approaches 0. So, statement IV is true.
Conclusion Therefore, statements I and IV are true.
Examples
Exponential functions are used to model population growth. If a population starts at 100 and grows by 5% each year, the population after x years can be modeled by y = 100 ( 1.05 ) x . Understanding the properties of exponential functions helps us predict how the population will change over time. For example, we know the population will always increase since the base is greater than 1.
