Graph y=e^(-x^2)

Problem

y=e(−x2)

Solution

1. Identify the function type and its domain. The function y=e(−x2) is a Gaussian function (a bell curve). The domain is all real numbers, (−∞,∞)

2. Determine symmetry by checking if the function is even or odd. Since ƒ*(−x)=e(−(−x)2)=e(−x2)=ƒ(x) the function is even, meaning the graph is symmetric about the yaxis.

3. Find the intercepts. To find the yintercept, set x=0 which gives y=e0=1 There are no xintercepts because e(−x2) is always greater than 0

4. Determine horizontal asymptotes by taking the limit as x approaches infinity. As x→∞ or x→−∞ the exponent −x2→−∞ so y→0 The xaxis (y=0 is a horizontal asymptote.

5. Find the first derivative to locate critical points. Using the chain rule:

d(y)/d(x)=−2*x*e(−x2)

Setting the derivative to 0 gives x=0 Since the derivative changes from positive to negative at x=0 there is a local maximum at (0,1)

6. Find the second derivative to determine concavity and inflection points. Using the product rule:

d2(y)/(d(x)2)=(4*x2−2)*e(−x2)

Setting the second derivative to 0 gives 4*x2−2=0 which results in x=±1/√(,2) These are the points of inflection where the graph changes concavity.

7. Sketch the graph based on these features. The graph starts near y=0 rises to a peak at (0,1) and falls back toward y=0 maintaining symmetry across the yaxis.

Final Answer

y=e(−x2)

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