Question

Sketch or computer plot a graph of the function \(y=e^{-x^{2}}.\)

Short answer

Plot key points: (0, 1), (1, 0.3679), (-1, 0.3679), (2, 0.0183), (-2, 0.0183). Connect points smoothly.

Step-by-step solution

- Understand the Function

The given function is an exponential function of the form \(y = e^{-x^2}\). The function decays very quickly as \(x\) moves away from 0. This is because the exponent \(-x^2\) becomes more negative as \(x\) increases or decreases, causing \(e^{-x^2}\) to approach zero.

- Identify Key Points

To sketch the graph, identify key points of the function. Calculate the value of the function at several key points along the x-axis. For example, evaluate \(y = e^{-x^2}\) at \(x = -2, -1, 0, 1, 2\).\(y(0) = e^{0} = 1\)\(y(1) = e^{-1} \approx 0.3679\)\(y(-1) = e^{-1} \approx 0.3679\)\(y(2) = e^{-4} \approx 0.0183\)\(y(-2) = e^{-4} \approx 0.0183\)

- Sketch the Axis and Points

Draw the x-axis and y-axis on graph paper or in a graphing software tool. Plot the key points identified in Step 2.(0, 1), (1, 0.3679), (-1, 0.3679), (2, 0.0183), (-2, 0.0183)

- Draw the Curve

Connect the points smoothly, creating a curve that approaches zero as x moves away from zero in both the positive and negative directions. The curve should be symmetric about the y-axis because \(y = e^{-x^2}\) is an even function. The peak of the curve is at \(y(0) = 1\).

- Describe Additional Features

Notice the bell shape of the graph. The function approaches 0 as \(x\) tends towards infinity or negative infinity, indicating that the tails of the bell curve get very close to, but never quite touch, the x-axis.

Key concepts

Exponential Decay

Exponential decay refers to the rapid decrease of a function's value as the input increases in magnitude. In the function given by the exercise, \(y = e^{-x^2}\), exponential decay is evident. As \(x\) moves away from 0 in either direction, the exponent \(-x^2\) becomes more negative. Because \(e\) raised to a very negative number approaches zero, the function value \(y\) quickly decays towards zero. This creates a curve that dips rapidly to near-zero values, showing a sharp decline on either side of the peak.

Key Points Identification

Identifying key points is a fundamental step in graphing any function. It involves evaluating the function at several significant points along the x-axis. For the function \(y = e^{-x^2}\), key points like \(x = -2, -1, 0, 1,\) and \(2\) help in understanding its behavior:

• \(y(0) = e^0 = 1\): The function reaches its maximum value.
• \(y(1) ≈ 0.3679\): The value at \(x = 1\).
• \(y(-1) ≈ 0.3679\): The value at \(x = -1\).
• \(y(2) ≈ 0.0183\): The value at \(x = 2\).
• \(y(-2) ≈ 0.0183\): The value at \(x = -2\).

These points give a clear idea of where to plot important markers on the graph and shape the curve.

Symmetry in Functions

Symmetry in functions simplifies the graphing process as it reveals inherent properties. The function \(y = e^{-x^2}\) is symmetric about the y-axis. This means that \(f(x) = f(-x)\) for any value of \(x\), making it an even function. This symmetry tells us that the graph on the left side of the y-axis mirrors the graph on the right. For instance, the values at \(x = 1\) and \(x = -1\) are equal, as are the values at \(x = 2\) and \(x = -2\). This mirror image feature helps in sketching a precise and balanced graph.

Sketching Graphs

Sketching graphs involves drawing the coordinate axes, plotting key points, and connecting them smoothly. For the function \(y = e^{-x^2}\), first draw the x-axis and y-axis. Then, plot the key points: \( (0, 1), (1, 0.3679), (-1, 0.3679), (2, 0.0183), (-2, 0.0183)\). Connect these points with a smooth curve that peaks at \(y = 1\) when \(x = 0\) and approaches the x-axis as \(x\) moves further from zero. The resulting curve will have a bell shape and will show the characteristics of exponential decay.

Even Functions

Even functions, characterized by \(f(x) = f(-x)\), display symmetry about the y-axis. The function \(y = e^{-x^2}\) is an example of an even function. This characteristic means that for every positive input \(x\), there is a negative input \(-x\) that gives the same output \(y\). This symmetry ensures that the left and right sides of the function's graph are identical, aiding in easier plotting and understanding of the function's overall shape.