Question

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 y-intercept.

c. it determines the initial value.

d. it determines the vertical shift.

Explanation:

To determine the role of the base \( b \) in the exponential function \( f(x)=a\cdot b^{x} \):

• For option b: The y - intercept of a function \( y = f(x) \) is found by setting \( x = 0 \). For \( f(x)=a\cdot b^{x} \), when \( x = 0 \), \( f(0)=a\cdot b^{0}=a\cdot1 = a \). So the y - intercept is determined by \( a \), not \( b \).
• For option c: The initial value of the exponential function (the value when \( x = 0 \)) is \( f(0)=a \), so it is determined by \( a \), not \( b \).
• For option d: The function \( f(x)=a\cdot b^{x} \) has no vertical shift in its basic form. A vertical shift would be of the form \( f(x)=a\cdot b^{x}+k \), where \( k \) determines the vertical shift. So \( b \) does not determine the vertical shift.
• For option a: In the exponential function \( f(x)=a\cdot b^{x} \), if \( b>1 \), the function is an exponential growth function (the function values increase as \( x \) increases). If \( 0 < b<1 \), the function is an exponential decay function (the function values decrease as \( x \) increases). So the base \( b \) determines whether the function is growing or decaying.

Answer:

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