Question

how can you identify exponential decay on a graph? a curve that oscillates between two values as x increases. a straight line that remains constant as x increases. a curve that rises steeply as x increases, indicating rapid increase. a curve that falls steeply as x increases, indicating rapid decrease.

Explanation:

To identify exponential decay on a graph, we recall the characteristics of an exponential decay function. An exponential decay function has the form \( y = ab^x \) where \( 0 < b < 1 \). As \( x \) (the independent variable, usually on the horizontal axis) increases, the value of \( y \) (the dependent variable, on the vertical axis) decreases, and it does so in a way that the rate of decrease is rapid at first and then may slow down but still follows the exponential pattern. Looking at the options:

• The first option describes a curve oscillating between two values (more like a periodic function, not exponential decay).
• The second option describes a horizontal line (constant function, not exponential decay).
• The third option describes a curve that rises steeply as \( x \) increases (this is exponential growth, not decay).
• The fourth option describes a curve that falls steeply as \( x \) increases, indicating rapid decrease. This matches the behavior of an exponential decay function, where as \( x \) increases, \( y \) decreases exponentially (rapidly at first, following the \( y=ab^x, 0 < b < 1 \) form).

Answer:

A curve that falls steeply as x increases, indicating rapid decrease.