Question

Sketch the graph of the function.\(y=e^{-0.1 x}\)

Short answer

The graph of the function \(y=e^{-0.1 x}\) is a curve that starts at a high point on the left, passes through the point (0,1), and approaches but never reaches the x-axis as it continues to the right. The exact shape depends on the calculated points: (0,1), (-1,1.105), (1,0.905), (2,0.819), (3,0.741).

Step-by-step solution

Identify the form of the function

The function given is of the form \(y=e^{-kx}\) where \(k=0.1\). This is a commonly known exponential decay function, so the graph decrease as x increases.

Calculate y-values for some significant x-values

Select a few x-values. For this type of function, it's generally good to include \(x=0\) (since any number to the 0 power is 1, so \(y=e^{-0.1*0}=1\)), and a couple of positive and negative x-values. For example, let's choose \(x=-1, 1, 2, 3\) and calculate the corresponding y-values: \(y=e^{-0.1*(-1)}=e^{0.1}\approx 1.105, y=e^{-0.1*1}=e^{-0.1}\approx 0.905, y=e^{-0.1*2}=e^{-0.2}\approx 0.819, y=e^{-0.1*3}=e^{-0.3}\approx 0.741.\)

Plot the points and sketch the graph

Plot the points you calculated on the coordinate plane: (0,1), (-1,1.105), (1,0.905), (2,0.819), (3,0.741). Notice that as \(x\) increases, the value of \(y\) decreases, but it never reaches 0. That's why the x-axis is a horizontal asymptote for the function. Connect the points with a smooth curve to create the graph of the function. The graph should show the typical pattern of exponential decay: it rises steeply to the left and levels out to the right.

Key concepts

Graph Sketching

Graph sketching is a powerful way to visualize complex functions like exponential decay. When you sketch a graph, you're essentially representing mathematical relationships in a visual format. This makes it easier to understand how variables interact with one another. To begin graph sketching, it's helpful to calculate a few key points, which are pairs of x-values and their corresponding y-values. For the function given, \(y = e^{-0.1x}\), a good approach is to pick a range of x-values both negative and positive to observe how the function behaves in those regions.

• Select x-values like \(x = -1, 0, 1, 2, 3\) to see the transition of the curve.
• Calculate the y-values for these x-values, such as \(y = e^{-0.1(0)} = 1\), illustrating that any number raised to the zero power equals one.
• Once you have all your points, plot them on your graph to see the trend.

Graph sketching becomes intuitive with practice, bringing to life the abstract concepts of mathematics. The graph of this function will display a downward trend, showing the nature of exponential decay. It's vital to remember that in sketching, perfection is not the goal, but rather grasping the function's behavior.

Coordinate Plane

The coordinate plane is the fundamental space where mathematical functions are visualized. It's a two-dimensional plane formed by the intersection of two lines, the horizontal x-axis and the vertical y-axis. Each point on this plane is determined by an ordered pair \((x, y)\), which denotes its position.

For the function \(y = e^{-0.1x}\), the coordinate plane allows you to plot points like \((0, 1)\), \((-1, 1.105)\), \((1, 0.905)\), and see their relationship visually. This interaction helps in understanding the function's behavior as x changes. To efficiently use the plane:

• Label your axes to clearly mark which is which.
• Use equal intervals along the axes to maintain consistent scaling.
• Plot the calculated points accurately to reflect the real behavior of the function.

This method helps in visualizing where a function increases, decreases, and how it trends towards an asymptote. The coordinate plane is thus essential for translating mathematical equations into visual representations that are easier to analyze and interpret.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches but never quite reaches. For the function \(y = e^{-0.1x}\), it's crucial to understand that x-axis serves as a horizontal asymptote. As \(x\) increases or decreases, \(y\) will get closer and closer to zero but never actually reach it.

This happens because an exponential decay function like \(y = e^{-kx}\), where \(k\) is a positive constant, decreases rapidly at first but then flattens out. Over infinite x-values, the function approaches the line \(y = 0\), but since the exponent will never achieve a negative infinity or zero, the y-value never fully reaches zero.

• As \(x\) gets very large, \(e^{-0.1x}\) becomes a very small positive number, close to zero.
• This behavior is depicted in the graph as it seemingly "hugs" the x-axis.
• The function thus visually represents its action of approaching the asymptote without contacting it.

Recognizing horizontal asymptotes helps in understanding the long-term behavior of functions, especially in calculus and analysis, where they provide insights into limits and end-behaviors of graphs.