Plot the Graph of y=xe^x how we draw these like graphs

To graph the function \(y = xe^x\), we can break down the process into manageable steps. This function is interesting because it combines polynomial and exponential behaviors, which leads to a unique shape. Let's go through how to effectively plot this graph.

Understanding the Function

The function \(y = xe^x\) consists of two parts: the polynomial term \(x\) and the exponential term \(e^x\). The exponential function grows very rapidly compared to the linear term as \(x\) increases. This means that our graph will have specific characteristics based on the interaction between these two components.

Finding Key Points

Before we plot the graph, we should identify some key points. These points can help us understand the shape of the graph more clearly. Here’s how we can find them:

• Evaluate at x = 0:

  When \(x = 0\), \(y = 0 \cdot e^0 = 0\). So, the point (0, 0) is on the graph.

• Evaluate at positive x-values:

  For \(x = 1\), \(y = 1 \cdot e^1 \approx 2.718\), giving us the point (1, 2.718).

  For \(x = 2\), \(y = 2 \cdot e^2 \approx 14.778\), leading to (2, 14.778).

• Evaluate at negative x-values:

  For \(x = -1\), \(y = -1 \cdot e^{-1} \approx -0.368\), which gives the point (-1, -0.368).

  For \(x = -2\), \(y = -2 \cdot e^{-2} \approx -0.270\), resulting in (-2, -0.270).

Analyzing Behavior at Extremes

Next, it’s useful to consider the behavior of the function as \(x\) approaches positive and negative infinity:

• As x approaches positive infinity:

  The term \(e^x\) dominates, so \(y\) will grow without bound.

• As x approaches negative infinity:

  The exponential term \(e^x\) approaches zero, and since \(x\) is negative, \(y\) will also approach zero from the negative side.

Graphing the Function

Now that we have some key points and an understanding of the function’s behavior, we can start plotting:

• Begin by marking the point (0, 0) on the graph.
• Next, plot the points we calculated: (1, 2.718), (2, 14.778), (-1, -0.368), and (-2, -0.270).
• Draw a smooth curve through these points, ensuring that the curve rises steeply as it goes to the right and approaches zero from below as it goes to the left.

Final Touches

Finally, label your axes with appropriate scales. The y-axis will need to accommodate very large values as \(x\) increases, and the x-axis should include both positive and negative values to show the behavior of the function in both directions. If you have graphing software or a graphing calculator, you can also visualize this function more precisely.

This process not only helps in plotting \(y = xe^x\) but also gives insight into how polynomial and exponential functions interact. Understanding these elements will aid you in graphing similar functions in the future!