Question

1. how does the base of an exponential function influence its graph?

a. it decides the x-intercept.
b. it controls the y-intercept.
c. it determines the growth or decay rate.
d. it affects the direction the graph opens.

Explanation:

• For option a: Exponential functions of the form \( y = a^x \) (where \( a>0,a

eq1 \)) never cross the x - axis (since \( a^x>0 \) for all real x), so the base does not decide the x - intercept.

• For option b: The y - intercept of \( y = a^x \) is found by setting \( x = 0 \), giving \( y=a^0 = 1 \) (regardless of the base \( a>0,a

eq1 \)), so the base does not control the y - intercept.

• For option c: If the base \( a>1 \), the exponential function \( y = a^x \) is a growth function (increasing as x increases). If \( 0 < a<1 \), the function \( y=a^x \) is a decay function (decreasing as x increases). So the base determines the growth or decay rate.
• For option d: Exponential functions do not "open" in a direction like parabolas (quadratic functions) do. Their graphs are either increasing (for \( a > 1\)) or decreasing (for \( 0 < a<1\)) curves, and the base does not affect a non - existent "opening direction".

Answer:

c. It determines the growth or decay rate