Question

how can you identify exponential growth on a graph?
a curve that oscillates between two values as x increases.
a curve that rises steeply as x increases, indicating rapid growth.
a straight line that remains constant as x increases.
a curve that gradually flattens out as x increases.

Explanation:

To identify exponential growth on a graph, we analyze the shape of the curve:

• Oscillating between two values (first option) is periodic, not exponential.
• A straight line with constant slope (third option) is linear, not exponential.
• A curve that flattens (fourth option) is likely logistic or decreasing in growth rate, not exponential.
• Exponential growth is characterized by a curve that rises steeply as \( x \) increases, with the growth rate accelerating (second option), matching the nature of exponential functions (\( y = ab^x, b>1 \)) where the rate of increase itself increases.

Answer:

A curve that rises steeply as \( x \) increases, indicating rapid growth.