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In the world of mathematics, an exponential function like \( y = 2e^x \) can undergo various transformations. The original function \( y = e^x \) serves as a base, and changes to this function can alter the graph's appearance and behavior. When looking at \( y = 2e^x \), the transformation involved is a vertical stretch.

A vertical stretch occurs when a constant is multiplied by the entire function, which, in this case, is the coefficient 2. This transformation stretches the graph vertically by a factor of 2.

So, compared to the base function \( y = e^x \), the graph of \( y = 2e^x \) is steeper and grows faster as \( x \) increases. It's as if you're pulling the graph upwards, making it "taller," but not changing its basic shape.

Exponential growth is a common characteristic of exponential functions such as \( y = 2e^x \). Understanding this concept is crucial for interpreting how the function behaves as \( x \) changes.

In exponential growth, the value of \( y \) increases at an increasing rate as \( x \) increases. For \( y = 2e^x \), since \( e^x \) itself is inherently an increasing function, multiplying it by 2 accelerates this growth.

• When \( x = 0 \), \( y = 2e^0 = 2 \).
• As \( x \) becomes positive, \( y \) increases rapidly—for instance, \( y \approx 5.436 \) at \( x = 1 \), and \( y \approx 14.872 \) at \( x = 2 \).

• Conversely, as \( x \) becomes negative, \( y \) approaches zero, since \( e^{-x} \) yields small values.

This behavior illustrates exponential growth, where growth is not linear, but quickens as values of \( x \) get larger.

Plotting graphs accurately is essential for visualizing mathematical functions such as \( y = 2e^x \). The process involves a few basic steps to ensure clarity and correctness.

Start by determining key points by substituting several \( x \) values into the function. For \( y = 2e^x \):

• For \( x = -1 \), \( y \approx 0.736 \).
• For \( x = 0 \), \( y = 2 \).
• For \( x = 1 \), \( y \approx 5.436 \).
• For \( x = 2 \), \( y \approx 14.872 \).

Plot these points on a graph, using a consistent scale to ensure accuracy. Then, draw a smooth curve that passes through each point.

As you sketch, observe the curve's trends: it should rise swiftly in the positive \( x \) direction and approach zero as \( x \) becomes negative. This visualization shows the exponential nature of the function, leading to insightful conclusions about its growth pattern.