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Exponential decay is a type of exponential function where the value decreases over time. It's a key characteristic because the base, in this case, is a fraction (1/2) which is less than one. When the exponent increases, the function value decreases gradually. This is observed in functions like radioactive decay, cooling processes, and depreciation.

• An exponential decay function can be identified by its formula: \( y = a^x \), where \( a < 1 \).
• The base value determines if the function is growing or decaying.

Understanding exponential decay is important because it shows us how quantities reduce quickly at first and then more slowly over time. When \( x \) becomes large, the function value gets closer to zero but never touches it. This indicates an asymptote at \( y = 0 \), meaning the graph will approach this line but will not cross or touch it.

Graph sketching involves plotting given points and understanding how the graph behaves on a coordinate plane. For exponential decay, such as our function \( y = (\frac{1}{2})^x \), this means identifying key points and drawing a smooth curve.

• Start by plotting points based on simple values of \( x \). For example, when \( x = 0, 1, -1 \).
• These points will give an idea of how the graph is shaped.

For \( y = (\frac{1}{2})^x \), crucial points include:- \( (0, 1) \): The base case showing initial value.- \( (1, \frac{1}{2}) \): Demonstrating the decay.- \( (-1, 2) \): Showing increase when \( x \) is negative.After plotting these points, draw a smooth curve through them starting from the left (where \( y \) is high when \( x \) is negative), passing through the plotted points and moving closer to the x-axis as \( x \) increases.

Function analysis is about understanding different features of the function. In our example, \( y = (\frac{1}{2})^x \), we can analyze it to obtain insights into its behavior, domain and range.

• The domain of exponential functions like this one is all real numbers \( x \), meaning \( x \) can be any value.
• The range, however, is all positive real numbers \( y > 0 \), as the function never touches or crosses the x-axis.

Additionally, identifying horizontal asymptotes helps in analysis. Here, \( y = 0 \) acts as an asymptote. The curve of the graph will get infinitely close to it but never actually reach it.

• Understanding key points like the y-intercept provides insights into where the curve starts. For our function, the y-intercept is at \( (0, 1) \). This is where the curve crosses the y-axis.
• The function decreases in value, indicating a continuous reduction in the y-value as \( x \) increases.

In summary, function analysis helps us break down the behavior of the function, making it easier to predict and understand its graph.