The family of curves represented by the equation y = e^(kx) is characterized by:

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Question: The family of curves represented by the equation y = e^(kx) is characterized by:

Options:

Correct Answer: Exponential growth

Solution:

The equation y = e^(kx) represents exponential growth for different values of k.

The family of curves represented by the equation y = e^(kx) is characterized by:

The family of curves represented by the equation y = e^(kx) is characterized by:

Step 1: Understand the equation y = e^(kx). This means y is equal to the number e (approximately 2.718) raised to the power of k times x.
Step 2: Recognize that 'k' is a constant that can change. It affects the shape of the curve.
Step 3: If k is positive, the curve will rise steeply as x increases, showing exponential growth.
Step 4: If k is negative, the curve will fall as x increases, showing exponential decay.
Step 5: Therefore, the family of curves represented by this equation can show different behaviors (growth or decay) depending on the value of k.

Exponential Functions – The equation y = e^(kx) represents a family of exponential functions where the base is Euler's number e and k determines the growth rate.
Growth and Decay – The value of k indicates whether the function represents exponential growth (k > 0) or decay (k < 0).