Problem 25 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph the function. $$ y=-2^{x} $$

Step-by-Step Solution

Answer

The graph of the function \(y=-2^{x}\) is an exponential graph reflected over the x-axis. It decreases rapidly for positive x and slowly approaches 0 for negative x, crossing the y-axis at the point (0,-1).

Step 1: Input the Function into the Graphing Utility

First, enter the provided function \(y=-2^{x}\) in the input field of the graphing utility.

Step 2: Adjust the View Settings

Most graphing programs start with a default view that might not show your curve completely. Adjust the x and y-axis values in the utility so that the graph of the function is clearly visible, typically a reasonable range for both x and y could be -10 to 10. Remember, for exponential functions the y-values can increase or decrease very rapidly.

Step 3: Interpret the Graph

The graph displays the general characteristics of an exponential function. The function \(y=-2^{x}\) will look like a regular 2^x graph but reflected over the x-axis because of the negative sign before the 2. The graph should decrease rapidly for positive x, and slowly approach 0 for negative x. It crosses the y-axis at (0,-1), because any number raised to the power of zero is 1 and there's a negative sign in front.

Key Concepts

Graphing Utility

A graphing utility is an invaluable tool for visualizing mathematical functions. These programs allow you to input mathematical equations and see their graphs almost instantly. This visual aid makes it much easier to understand complex functions like exponential ones. Here's how you can maximize the use of a graphing utility:

• First, make sure to correctly input your function. This is crucial as even a small mistake in the equation could lead to an incorrect graph.
• Experiment with the view settings. Often, you might need to adjust the x and y-axis range to get a clear picture of the function's behavior.
• Look for features like zoom and trace to explore different parts of the graph in more detail. These features can help you better understand key aspects like intercepts and asymptotes.

Understanding how to use these utilities effectively will greatly enhance your ability to analyze mathematical graphs.

Function Reflection

A function reflection is an interesting transformation in mathematics that flips the graph of a function over one of the axes. For instance, when dealing with the function \(y = -2^x\), the graph reflects the function \(y = 2^x\) over the x-axis. This is due to the negative sign in front of the base.When a function is reflected over the x-axis:

• The positive y-values become negative, and the negative y-values remain negative but are mirrored downwards.
• This transformation doesn't alter the x-values, which means the function's horizontal alignment remains intact.

The resulting graph shows the flipped behavior, where the curve that typically rises quickly for positive x-values now decreases rapidly instead.

Exponential Decay

Exponential decay describes a process where quantities decrease rapidly at a rate proportional to their current value. In mathematical terms, it is modeled by functions like \(y = -b^x\) where \(0 < b < 1\) or in our case \(y = -2^x\), but the negative sign causes the decay.Key characteristics of exponential decay include:

• The curve approaches the x-axis but never touches it. This characteristic signifies that the function gets closer and closer to zero without ever actually being zero.
• In the given function, for positive x-values, the exponential expression decreases as the x-value increases. This creates a steep decline on the graph.
• Exponential decay functions have a rapid start where values drop quickly, followed by a slow approach to zero as they level out.

Understanding exponential decay through its graph helps in visualizing how quantities diminish over time, which is particularly useful in fields like physics and finance.