Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=2+\left(x^{2}-3\right) e^{-x} $$

Short answer

The graph of the function \(f(x) = 2 + (x^2 - 3)e^{-x}\) reveals its extrema and asymptotes. After careful observation of the graph, these values are identified and labeled accordingly.

Step-by-step solution

Graph the Function

Input the function \(f(x) = 2 + (x^2 - 3)e^{-x}\) into a computer algebra system or graphing calculator to generate the graph. Ensure to have an adequate view of the function's behaviors at larger or smaller x-values as well.

Identify the Extrema

Analyze the graph for any high or low points which are local maxima or minima. Check the points where the slope of the graph changes from positive to negative (maxima) or from negative to positive (minima).

Analyze the Asymptotes

Investigate the graph for any horizontal or vertical asymptotes. A horizontal asymptote is a horizontal line that the graph approaches as x approaches positive or negative infinity, while a vertical asymptote is a vertical line that the graph approaches as x approaches a specific value. Note down these values for x if any exist.

Key concepts

Extrema on Graphs

Understanding the extrema on graphs is crucial in graph analysis, as it involves identifying the highest and lowest points, known as the maximum and minimum points, respectively. An extremum can be 'local', meaning it is the highest or lowest point in a nearby region of the graph, or 'global', which stands as the absolute highest or lowest across the entire graph.

To properly identify extrema using a computer algebra system, follow these guidelines: First, graph the function to get an overall view. Then look for points where the graph changes direction, indicating potential extrema. In the context of the function f(x) = 2 + (x^2 - 3)e^{-x}, one would need to find the derivative of the function and determine where it is zero or does not exist. These points, known as critical points, are potential locations for extrema. By examining the behavior of the function around these points, you can classify them as either maxima or minima. Use visualization tools provided by the computer algebra system to confirm these findings, looking for peaks and troughs on the graph.

Asymptotes Identification

Asymptotes are lines that a graph approaches as the inputs or outputs reach infinity, serving as boundaries that the graph will never quite touch. Identifying them correctly is an important aspect of graph analysis.

There are three types of asymptotes: horizontal, vertical, and oblique (slant). Horizontal asymptotes occur if the value of the function approaches a constant as x approaches infinity or negative infinity. For the given function f(x) = 2 + (x^2 - 3)e^{-x}, you would examine the behavior of f(x) as x goes to positive or negative infinity. Here, e^{-x} tends to zero as x goes to infinity, suggesting a horizontal asymptote at y=2.

Vertical asymptotes are found where the function becomes undefined, typically at points of division by zero in rational functions. However, for exponential functions like the given one, vertical asymptotes are uncommon. Lastly, oblique asymptotes may occur when the degree of the numerator is exactly one greater than the degree of the denominator in a rational function, which is not the case here. When using a computer algebra system, graphical representation will aid in observing these asymptotic behaviors, where the graph seems to 'flatten out' or shoot off infinitely.

Exponential Functions Graphing

Graphing exponential functions often reveals their rapid growth or decay, and these functions are characterized by a constant base raised to a variable exponent. When graphing exponential functions, it's important to note features like intercepts, increases or decreases, and especially the horizontal asymptote, which is common in these types of functions.

The exponential function in our exercise, f(x) = 2 + (x^2 - 3)e^{-x}, is a mix of polynomial and exponential expressions. Here, the e^{-x} term signifies exponential decay, as e is raised to a negative power of x. When graphing, we notice that this component diminishes as x becomes large, leading the polynomial term x^2 to dominate the behaviour for large negative x.

To graph exponential functions effectively with a computer algebra system, ensure that the viewing window includes a sufficient range of x values to capture the decay or growth. Pay close attention to the behavior as x approaches large positive or negative values to identify the horizontal asymptote accurately, and to observe how the exponential term affects the function's overall shape. With a function that combines polynomial and exponential terms, observe interaction between these different behaviours to fully understand the graph's dynamics.