Question

1. which of the following best describes the role of the base ( b ) in the function ( f(x) = a cdot b^x )?

a. it determines whether the function is growing or decaying.

b. it determines the initial value.

c. it determines the ( y )-intercept.

d. it determines the vertical shift.

Explanation:

• For option a: In the exponential function \( f(x)=a\cdot b^{x} \), if \( b > 1 \), the function is growing; if \( 0 < b < 1 \), the function is decaying. So the base \( b \) determines if the function grows or decays.
• For option b: The initial value (when \( x = 0 \)) is \( a\cdot b^{0}=a \), so the initial value is determined by \( a \), not \( b \).
• For option c: The y - intercept occurs at \( x = 0 \), and \( f(0)=a\cdot b^{0}=a \), so the y - intercept is determined by \( a \), not \( b \).
• For option d: There is no vertical shift in the function \( f(x)=a\cdot b^{x} \) (a vertical shift would be of the form \( f(x)=a\cdot b^{x}+k \), where \( k \) is the vertical shift). So \( b \) does not determine a vertical shift.

Answer:

a. It determines whether the function is growing or decaying.