Question

which properties are present in a table that represents an exponential function in the form (y = b^x) when (b > 1)?
i. as the (x)-values increase, the (y)-values increase.
ii. the point ((1, 0)) exists in the table.
iii. as the (x)-values increase, the (y)-values decrease.
iv. as the (x)-values decrease, the (y)-values decrease, approaching a singular value.

(a) i and iv
(b) i and ii
(c) ii and iii
(d) iii only

Explanation:

Analyze growth behavior

Using the Exponential Growth and Decay Behavior knowledge point
Since \(b > 1\), the function \(y = b^x\) represents exponential growth. As \(x\) increases, \(y\) increases. Thus, statement I is true, and statement III is false.

Evaluate specific points

Using the Exponential Function Properties knowledge point
Substituting \(x = 1\) into \(y = b^x\) yields \(y = b^1 = b\). Since \(b > 1\), the point on the graph is \((1, b)\), not \((1, 0)\). Thus, statement II is false.

Analyze asymptotic behavior

Using the Exponential Function Properties knowledge point
As \(x\) decreases toward negative infinity (\(x \to -\infty\)), \(y = b^x = \frac{1}{b^{-x}}\) decreases and approaches \(0\). Thus, statement IV is true.

Combine true statements

Combining the results, statements I and IV are the correct properties present in the table.

Answer:

• I and IV (Correct answer)
• I and II
• II and III
• III only