What is the family of curves represented by the equation y = e^(kx)?

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

Options:

Correct Answer: Exponential functions with varying growth rates

Solution:

The equation y = e^(kx) represents a family of exponential functions where \'k\' determines the growth rate.

What is the family of curves represented by the equation y = e^(kx)?

What is the family of curves represented by the equation y = e^(kx)?

Correct Answer: y = e^(kx) represents a family of exponential functions.

Step 1: Understand the equation y = e^(kx). Here, 'e' is a constant (approximately 2.718), and 'k' is a variable that can change.
Step 2: Recognize that 'k' affects the shape of the curve. If 'k' is positive, the curve will rise as x increases (exponential growth).
Step 3: If 'k' is negative, the curve will fall as x increases (exponential decay).
Step 4: Realize that different values of 'k' create different curves, but they all share the same basic shape (exponential).
Step 5: Conclude that the equation y = e^(kx) represents a family of curves, each defined by a specific value of 'k'.

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