91Ó°ÊÓ

In graphical terms, the y-intercept is the point where the graph of a function crosses the y-axis. To find the y-intercept of an exponential function such as \( y = -5(1.6)^{-x} \), we set \( x \) to 0 because the y-intercept is always where \( x = 0 \).
Plugging in \( x = 0 \), we get:

• \( y = -5(1.6)^0 \)
• \( y = -5 \times 1 \)
• \( y = -5 \)

Thus, the y-intercept is the point \( (0, -5) \), meaning the graph will cross the y-axis at \( y = -5 \). This crucial point helps to anchor the position of the curve on the coordinate plane, providing a starting point for graphing the function.

A horizontal asymptote represents a line that the graph of the function approaches as \( x \) heads towards infinity or negative infinity, but never actually reaches. For the function \( y = -5(1.6)^{-x} \), as \( x \to \infty \), \( (1.6)^{-x} \) approaches 0, and so does \( y \).
This suggests the presence of a horizontal asymptote at \( y = 0 \). It's essentially a guide that tells us the long-term behavior of the function. Exponential functions tend to "flatten out" as they extend far to the right on a graph, approaching but never touching this imaginary boundary. Understanding the horizontal asymptote gives insight into how exponential functions behave with large or small values of \( x \).

Plotting on a coordinate plane means placing points on a grid that pair \( x \) values with corresponding \( y \) values as calculated from the function. These points are then connected to display the graph's overall shape.

• For example, for \( x = -2 \), \( y = -12.8 \) making the point \( (-2, -12.8) \).
• For \( x = 0 \), \( y = -5 \) making the point \( (0, -5) \).
• For \( x = 2 \), \( y = -1.953 \) making the point \( (2, -1.953) \).

When these points are plotted and connected on the plane, a smooth curve emerges, outlining the underlying pattern of the exponential function. This visual guide is critical in conveying how the exponential function operates over a range of values.

Exponential decay occurs when a quantity diminishes at a consistent percentage rate over time, resulting in a rapid drop-off. In the function\( y = -5(1.6)^{-x} \), the negative exponent causes the base \( 1.6 \) to decrease because \( (1.6)^{-x} \) is equivalent to \( \frac{1}{(1.6)^x} \).
Hence, as \( x \) increases, the function value declines, demonstrating exponential decay.

• The multiplier \(-5\) further amplifies this decrease, flipping the graph upside-down.
• This creates a graph that starts high and descends towards zero as it moves from left to right.

Understanding exponential decay is important in fields like finance, science, and engineering, where it models processes that shrink over time, such as depreciating assets or cooling temperatures.