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The y-intercept is a fundamental concept in graphing functions. It's where a graph crosses the y-axis. For any function, to find the y-intercept, you simply set the x-value to 0 in the given equation and solve for y. In the function given, \( y = -2^x \), setting \( x = 0 \) yields \( y = -2^0 = -1 \). Therefore, the y-intercept of this exponential function is represented by the coordinate point \((0, -1)\).

This point not only gives us a starting point on the graph but also indicates the initial value of the function when \( x \) is zero. The importance of the y-intercept is considerable, especially when plotting the curve of exponential functions like \( y = -2^x \).

In the realm of exponential functions, the horizontal asymptote represents a line that the graph approaches but never actually touches. For functions of the form \( y = a \, imes \, b^x \), where \( b > 0 \), this line is often the x-axis, or \( y = 0 \).

In our specific function \( y = -2^x \), as \( x \) approaches negative infinity, the value of \( y \) gets closer and closer to zero. The horizontal asymptote in this case is \( y = 0 \). Even though the graph will approach this line infinitely, it will never intersect or touch it.

Grasping the concept of horizontal asymptotes is key for understanding the behavior of the function's graph, especially as it stretches into negative infinity.

Graphing exponential functions can initially seem daunting. However, breaking it down into smaller steps can ease the process. To graph a function such as \( y = -2^x \), you start by identifying key features like the y-intercept and the horizontal asymptote.

In this function:

• The y-intercept is at \( (0, -1) \).
• The horizontal asymptote is at \( y = 0 \).

With this information and a table of values, you can plot several specific points.

Next, draw a smooth curve by connecting these points, ensuring that it slopes downwards as \( x \) increases, showing how negative exponential functions behave. Specifically, as \( x \) grows, \( y \) values become more negative, moving away from the horizontal asymptote, yet getting closer as \( x \) diminishes.

A table of values is a valuable tool when graphing functions, especially exponential ones like \( y = -2^x \). By selecting specific \( x \)-values and calculating their corresponding \( y \)-values, we gain clear insights into the behavior of the function.

Here’s how it works in our function:
For \( x = -2, -1, 0, 1, \) and \( 2 \), the table shows:

• \((x, y) = (-2, 4)\)
• \((x, y) = (-1, 2)\)
• \((x, y) = (0, -1)\)
• \((x, y) = (1, -2)\)
• \((x, y) = (2, -4)\)

Using these values, you can plot points on a coordinate plane.

Constructing a table helps in creating an accurate graph and understanding how the values shift as \( x \) increases or decreases. This not only assists in better visualization but also reinforces the conceptual understanding of exponential functions and their characteristics.