Answer :
To determine which properties are present in a table that represents an exponential function in the form [tex]\( y = b^x \)[/tex] when [tex]\( b > 1 \)[/tex], we need to analyze what happens to the function under different scenarios. Here's the step-by-step reasoning:
1. Property I: As the [tex]\( x \)[/tex]-values increase, the [tex]\( y \)[/tex]-values increase.
- For an exponential function [tex]\( y = b^x \)[/tex] with [tex]\( b > 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] will also increase. This is because raising a number greater than 1 to a higher power results in a larger number.
2. Property II: The point [tex]\( (1, 0) \)[/tex] exists in the table.
- This property is incorrect. For the exponential function [tex]\( y = b^x \)[/tex], when [tex]\( x = 1 \)[/tex], [tex]\( y = b^1 = b \)[/tex], not 0. Therefore, the point [tex]\( (1, 0) \)[/tex] does not exist in such a table.
3. Property III: As the [tex]\( x \)[/tex]-values increase, the [tex]\( y \)[/tex]-values decrease.
- This property is also incorrect. As explained in Property I, for [tex]\( b > 1 \)[/tex], [tex]\( y \)[/tex] increases as [tex]\( x \)[/tex] increases.
4. Property IV: As the [tex]\( x \)[/tex]-values decrease, the [tex]\( y \)[/tex]-values decrease, approaching a singular value.
- This property is true. As [tex]\( x \)[/tex] decreases, especially towards negative infinity, [tex]\( y = b^x \)[/tex] where [tex]\( b > 1 \)[/tex] will approach zero. Thus, [tex]\( y \)[/tex] decreases and approaches a singular value (zero).
By analyzing these properties, we conclude that the correct properties are:
I and IV
1. Property I: As the [tex]\( x \)[/tex]-values increase, the [tex]\( y \)[/tex]-values increase.
- For an exponential function [tex]\( y = b^x \)[/tex] with [tex]\( b > 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] will also increase. This is because raising a number greater than 1 to a higher power results in a larger number.
2. Property II: The point [tex]\( (1, 0) \)[/tex] exists in the table.
- This property is incorrect. For the exponential function [tex]\( y = b^x \)[/tex], when [tex]\( x = 1 \)[/tex], [tex]\( y = b^1 = b \)[/tex], not 0. Therefore, the point [tex]\( (1, 0) \)[/tex] does not exist in such a table.
3. Property III: As the [tex]\( x \)[/tex]-values increase, the [tex]\( y \)[/tex]-values decrease.
- This property is also incorrect. As explained in Property I, for [tex]\( b > 1 \)[/tex], [tex]\( y \)[/tex] increases as [tex]\( x \)[/tex] increases.
4. Property IV: As the [tex]\( x \)[/tex]-values decrease, the [tex]\( y \)[/tex]-values decrease, approaching a singular value.
- This property is true. As [tex]\( x \)[/tex] decreases, especially towards negative infinity, [tex]\( y = b^x \)[/tex] where [tex]\( b > 1 \)[/tex] will approach zero. Thus, [tex]\( y \)[/tex] decreases and approaches a singular value (zero).
By analyzing these properties, we conclude that the correct properties are:
I and IV
