Problem 12 - Calculus for Biology and Medicine

Question

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\exp (-x) $$

Step-by-Step Solution

Answer

The graph is a decreasing exponential curve with a y-intercept at (0, 1) and a horizontal asymptote at y=0.

Step 1: Understanding the Function

The function given is \( y = \exp(-x) \), which is the exponential function with base \( e \) and a negative exponent. This characteristic leads to the graph having a certain behavior: as the exponent \( -x \) increases, the overall value of \( y \) decreases.

Step 2: Identify Domain and Range

For the function \( y = \exp(-x) \), the domain consists of all real numbers \( x \). This means you can substitute any real number into \( x \) and still have a defined output. The range of the function is all positive real numbers \( y > 0 \) because an exponential function always outputs a positive number.

Step 3: Analyze Intercepts

To find any intercepts, set \( x = 0 \). This gives \( y = \exp(0) = 1 \). Hence, the y-intercept is \( (0, 1) \). There is no x-intercept because the function never touches the x-axis; it's always positive.

Step 4: Determine Asymptotic Behavior

An asymptote is a line that the graph approaches but never touches. For \( y = \exp(-x) \), as \( x \to \, \infty \), the function approaches \( y = 0 \) but never quite gets there, so \( y = 0 \) (the x-axis) is a horizontal asymptote.

Step 5: Sketch the Graph

Use the information gathered to sketch the graph. It starts from the point \( (0, 1) \), rapidly decreases as \( x \) increases and gets closer but never reaches the x-axis as \( y = 0 \). As \( x \) becomes more negative, \( y \) increases exponentially, making the graph rise steeply to the left.

Key Concepts

Graph Sketching

Sketching the graph of exponential functions like \( y = \exp(-x) \) may sound tricky, but when broken down into parts, it becomes much simpler. The function itself is a classic example of an exponential decay graph because of its negative exponent. Let’s break it down step by step.

• Starting Point: The first important coordinate is the y-intercept where \( x = 0 \). At this point, \( y = \exp(0) = 1 \). This means the graph crosses the y-axis at (0, 1).
• Slope Direction: As \( x \) increases to the positive, the graph starts high and rapidly moves downward toward the x-axis.
• Rising Left: As \( x \) becomes negative, the graph actually rises. This is because the negative exponent becomes less negative, thus increasing \( y \) steeply.

With these traits, you can sketch the graph showing the quick drop off to the right and a steep upward rise to the left. These visual cues support seeing how the graph behaves without the need for computational graphing technology.

Asymptotes

Asymptotes are invisible lines that your function’s graph will get very close to, but never actually touch or cross. In the case of \( y = \exp(-x) \), understanding asymptotes helps you see the limits and bounds of a function. Here's what to look for:

• Horizontal Asymptote: The x-axis, or \( y = 0 \), acts as a horizontal asymptote for this function. As \( x \to \infty \), \( y \) approaches zero. It never quite reaches the x-axis, emphasizing the persistent positivity of the exponential function.
• Graph Behavior Near Asymptote: The graph descends toward this horizontal line but eternally floats just above it. A characteristic of decay functions is such asymptotic behavior, where the values of the function keep halving as they near the horizontal boundary.

Recognizing this aspect is key in sketching and understanding how exponential graphs forge their paths visually on a plot.

Function Behavior

Understanding the behavior of \( y = \exp(-x) \) is essential to grasp how exponential functions work, especially with negative exponents. Observably, this function expresses exponential decay. Let's break down the key behavioral traits:

• Decay: This function diminishes rapidly as \( x \) becomes positive. The higher the value of \( x \), the smaller \( y \) becomes, drawing close to zero without reaching it.
• No X-Intercepts: Notice that it never touches or crosses the x-axis. This reiterates the fact that the function's output never dips into negative numbers.
• Exponential Growth in Negative Direction: Interestingly, for negative \( x \), the function value rises. As \( x \) becomes more negative, the exponent decreases negatively, causing multiplication of a quickly enlarging number.

These notable behaviors highlight how the curve progresses logically with respect to its exponential decay nature, making it a fascinating subject in graph analysis.