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)
Property I is true because as x increases, y increases when b > 1.
Property II is false because the point (1, 0) does not exist on the graph of y = b x when b > 1.
Property III is false because as x increases, y increases when b > 1.
Property IV is true because as x decreases, y decreases, approaching 0. Therefore, the correct answer is I and IV. I and IV
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 are true for this function. Let's analyze each property.
Analyzing Property I Property I states: As the x -values increase, the y -values increase. Since 1"> b > 1 , as x increases, b x also increases. For example, if b = 2 , then as x goes from 1 to 2, y goes from 2 1 = 2 to 2 2 = 4 . So, Property I is true.
Analyzing Property II Property II states: 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 Property II is false.
Analyzing Property III Property III states: As the x -values increase, the y -values decrease. This is the opposite of what we found in Property I. Since 1"> b > 1 , as x increases, y increases, not decreases. Thus, Property III is false.
Analyzing Property IV Property IV states: As the x -values decrease, the y -values decrease, approaching a singular value. As x decreases (i.e., becomes more negative), b x decreases. As x approaches − ∞ , b x approaches 0. For example, if b = 2 , as x goes to − ∞ , y = 2 x approaches 0. Thus, Property IV is true.
Conclusion Therefore, Properties I and IV are present in the table representing the exponential function y = b x when 1"> b > 1 .
Examples
Exponential functions are used to model population growth. If a population starts at a size of 100 and grows by 5% each year, the population size after x years can be modeled by the function y = 100 ( 1.05 ) x . Since the base 1.05 is greater than 1, the population will increase over time, illustrating Property I. As time goes backwards (x becomes more negative), the population size approaches 0, illustrating Property IV.
