Answer :
To determine which properties are present in a table that represents an exponential function in the form \(y=b^x\) where \(b>1\), let's analyze each statement in detail:
Property I: As the \(x\)-values increase, the \(y\)-values increase.
For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) increases, \(b^x\) also increases because the base \(b\) raised to a higher power results in a larger number.
- Therefore, this property is valid.
Property II: The point \((1,0)\) exists in the table.
For an exponential function \(y = b^x\) with \(b > 1\):
- If \(x = 1\), then \(y = b^1 = b\), not 0.
- Therefore, this property is invalid because for any \(b > 1\), \(y\) will never be 0 when \(x = 1\).
Property III: As the \(x\)-values increase, the \(y\)-values decrease.
For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) increases, \(y = b^x\) increases as stated before.
- Therefore, this property is invalid because \(y\) does not decrease with increasing \(x\).
Property IV: As the \(x\)-values decrease, the \(y\)-values decrease, approaching a singular value.
For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) decreases, \(y = b^x\) decreases and approaches 0 as \(x\) approaches negative infinity.
- Therefore, this property is valid as \(y\) gets closer and closer to 0.
Thus, the valid properties for an exponential function \(y = b^x\) where \(b>1\) are:
- Property I: As the \(x\)-values increase, the \(y\)-values increase.
- Property IV: As the \(x\)-values decrease, the \(y\)-values decrease, approaching a singular value (0).
Therefore, the correct answer is:
I and IV
Property I: As the \(x\)-values increase, the \(y\)-values increase.
For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) increases, \(b^x\) also increases because the base \(b\) raised to a higher power results in a larger number.
- Therefore, this property is valid.
Property II: The point \((1,0)\) exists in the table.
For an exponential function \(y = b^x\) with \(b > 1\):
- If \(x = 1\), then \(y = b^1 = b\), not 0.
- Therefore, this property is invalid because for any \(b > 1\), \(y\) will never be 0 when \(x = 1\).
Property III: As the \(x\)-values increase, the \(y\)-values decrease.
For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) increases, \(y = b^x\) increases as stated before.
- Therefore, this property is invalid because \(y\) does not decrease with increasing \(x\).
Property IV: As the \(x\)-values decrease, the \(y\)-values decrease, approaching a singular value.
For an exponential function \(y = b^x\) with \(b > 1\):
- When \(x\) decreases, \(y = b^x\) decreases and approaches 0 as \(x\) approaches negative infinity.
- Therefore, this property is valid as \(y\) gets closer and closer to 0.
Thus, the valid properties for an exponential function \(y = b^x\) where \(b>1\) are:
- Property I: As the \(x\)-values increase, the \(y\)-values increase.
- Property IV: As the \(x\)-values decrease, the \(y\)-values decrease, approaching a singular value (0).
Therefore, the correct answer is:
I and IV
