Problem 32 - Elementary and Intermediate Algebra

Question

Graph each function. $$ f(x)=\frac{1}{2} e^{x} $$

Step-by-Step Solution

Answer

Identify key points, plot them, and draw a smooth exponential curve.

Step 1: Understand the function type

The function given is an exponential function of the form \( f(x) = ae^{x} \), where \( a = \frac{1}{2} \) in this case. Exponential functions grow at a rate proportional to their current value and produce a curve that increases exponentially.

Step 2: Determine the key characteristics

Identify the horizontal asymptote: This function has a horizontal asymptote at \( y = 0 \).Identify the y-intercept: The function will intersect the y-axis at \( x = 0 \). Substituting \( x = 0 \) into the equation gives \( f(0) = \frac{1}{2}e^{0} = \frac{1}{2} \). Thus, the y-intercept is at point (0, 0.5).

Step 3: Calculate a few function values

Choose some values for \( x \) to calculate specific points on the graph:- When \( x = -1 \), \( f(-1) = \frac{1}{2}e^{-1} \approx 0.184 \, (1/2e) \)- When \( x = 1 \), \( f(1) = \frac{1}{2}e^{1} \approx 1.359 \, (\approx 1.36) \)- When \( x = 2 \), \( f(2) = \frac{1}{2}e^{2} \approx 3.694 \, (\approx 3.69) \). Use these coordinates to help in plotting.

Step 4: Sketch the graph

Start by plotting the calculated points: (0, 0.5), (-1, 0.184), (1, 1.36), and (2, 3.69) on a set of axes. Draw a smooth curve through these points, noting the rapid increase as \( x \) becomes larger.Ensure to show that the curve approaches but does not touch the horizontal asymptote at \( y = 0 \) as \( x \) approaches negative infinity.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions can seem tricky at first, but understanding the basic steps can make it intuitive. Exponential functions like \( f(x) = \frac{1}{2} e^{x} \) have a distinctive curve. To start, identify the general form of the function, which is \( f(x) = ae^{x} \). Here, \( a = \frac{1}{2} \). The function describes exponential growth because the base, \( e \), is greater than 1. This means as \( x \) increases, \( f(x) \) grows rapidly.

To begin graphing, find key points by selecting values for \( x \). These can be easy-to-calculate numbers like \(-1, 0, 1, \) and \( 2 \). For instance, we can calculate:\

• \( f(0) = \frac{1}{2}e^{0} = \frac{1}{2} \)
• \( f(-1) = \frac{1}{2}e^{-1} \approx 0.184 \)
• \( f(1) = \frac{1}{2}e^{1} \approx 1.36 \)
• \( f(2) = \frac{1}{2}e^{2} \approx 3.69 \)

Once you have these points, plot them on a graph. Then, draw a smooth curve through them, ensuring it shows the exponential growth pattern.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches but never actually meets. In the context of exponential functions, it shows where the function levels off as \( x \) decreases or increases infinitely.

For the function \( f(x) = \frac{1}{2} e^{x} \), the horizontal asymptote is at \( y = 0 \). This occurs because as \( x \) approaches negative infinity, \( e^{x} \) gets extremely small, getting closer and closer to 0.

• The graph of \( f(x) = \frac{1}{2} e^{x} \) will approach \( y = 0 \) as \( x \to -\infty \), but it never reaches or crosses the line.
• The curve remains above the horizontal asymptote because exponentials are always positive.

Identifying horizontal asymptotes helps in knowing the behavior of the function as it stretches out across the axes.

Y-Intercept

The y-intercept of a function is the point where the graph intersects the y-axis. It gives us an initial point to start sketching the curve.

To find the y-intercept of the function \( f(x) = \frac{1}{2}e^{x} \), simply set \( x = 0 \). Substituting in gives:

• \( f(0) = \frac{1}{2}e^{0} = \frac{1}{2} \)

Therefore, the y-intercept is at the point \( (0, 0.5) \).

• This point is crucial as it serves as a starting point on your graph.
• The y-intercept tells us the value of the function at the very beginning of the curve.

Knowing how to find the y-intercept is an important skill because it provides a solid reference for plotting the overall trajectory of an exponential graph.