Graph y=e^x

Problem

y=ex

Solution

1. Identify the function type: The equation y=ex represents a basic exponential function where the base is Euler's number e≈2.718

2. Determine the y-intercept: Substitute x=0 into the equation to find the point where the graph crosses the vertical axis.

y=e0=1

The y-intercept is (0,1)

3. Analyze the horizontal asymptote: As x approaches negative infinity, the value of ex approaches 0

(lim_x→−∞)(ex)=0

The line y=0 (the x-axis) is a horizontal asymptote.

4. Determine the growth behavior: Since the base e is greater than 1 the function is strictly increasing. As x increases, y grows exponentially toward infinity.

(lim_x→∞)(ex)=∞

5. Calculate additional points: Evaluate the function at x=1 and x=−1 to help shape the curve.

x=1⇒y=e1≈2.718

x=−1⇒y=e(−1)≈0.368

6. Sketch the graph: Draw a smooth curve passing through (−1,0.368) (0,1) and (1,2.718) ensuring the curve stays above the x-axis and rises sharply to the right.

Final Answer

The graph of y=ex is an increasing exponential curve with a y-intercept at (0,1) and a horizontal asymptote at y=0

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