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Exponential decay describes a pattern where the value of a function decreases rapidly at first and then slows down over time. The function given in the exercise, \( y = e^{-x} \), is a classic example of exponential decay.

In this function, as \( x \) increases, the value of \( e^{-x} \) gets smaller. This happens because \( e \) (approximately 2.718) raised to a negative power means we are dealing with fractional values. Unlike linear decay, exponential decay causes the function to drop quickly and then slow down significantly.

Here are a few key points that help define the graph:

• When \( x = 0 \), \( y = e^{0} = 1 \).
• When \( x = 1 \), \( y = e^{-1} \approx 0.367 \).
• When \( x = 2 \), \( y = e^{-2} = \approx 0.135 \).

Notice the rapid decrease in the \( y \) values as \( x \) increases. This is the hallmark of exponential decay.

A horizontal asymptote is a horizontal line that a graph approaches but never actually touches. For the given function \( y = e^{-x} \), the horizontal asymptote is the x-axis, or \( y = 0 \).

Understanding horizontal asymptotes is crucial for sketching exponential functions. As \( x \) goes to infinity (i.e., becomes very large), the value of \( y \) approaches 0 but never actually reaches 0. This is because \( e^{-x} \) gets closer and closer to zero but remains positive. So, the curve will get infinitely close to the x-axis without touching it.

In summary, the key points to remember about horizontal asymptotes are:

• They represent a value that the function approaches but never meets.
• For \( y = e^{-x} \), the horizontal asymptote is \( y = 0 \).

This helps visualize the flattening of the curve as \( x \) increases.

The coordinate plane is the foundation for graphing functions like \( y = e^{-x} \). It consists of an x-axis (horizontal) and a y-axis (vertical) that intersect at the origin (0,0).

When graphing on a coordinate plane, you plot points based on their \( x \) and \( y \) values. For the exercise, some key points were:

• (0, 1) - When \( x = 0 \), \( y = 1 \).
• (1, 0.367) - When \( x = 1 \), \( y \approx 0.367 \).
• (2, 0.135) - When \( x = 2 \), \( y \approx 0.135 \).

After plotting these points, we join them with a smooth curve. This curve represents the function's behavior.

Key aspects to keep in mind when using a coordinate plane:

• Label the axes correctly to avoid mistakes.
• Ensure plotted points are accurate.
• Draw a smooth curve for functions like \( y = e^{-x} \).

By understanding the coordinate plane, you ensure that your graph accurately represents the function, making interpretation easier and clearer.