How do you graph #y=1/2e^x+1#?

Choose the "base function" to graph and then make the necessary adjustments for the modifiers.

graph{e^x [-5, 5, 5, 5, 5]}

graph{(1/2)e^x[-5,5,-5,5]}

Let's now lift the graph 1 unit upward to account for the +1 to complete the task:

graph{(1/2)e^x+1 [–5,–5,–5]}

To graph the function (y = \frac{1}{2}e^x + 1), follow these steps:

1. Identify key features of the function:

   • The base function is (e^x), which is an exponential function with a horizontal asymptote at (y = 0) and increasing behavior as (x) increases.
   • The function is vertically stretched by a factor of (1/2).
   • The entire graph is shifted vertically upward by 1 unit.

2. Plot key points:

   • When (x = 0), (y = \frac{1}{2}e^0 + 1 = \frac{1}{2} + 1 = \frac{3}{2}), so the point ((0, \frac{3}{2})) is on the graph.
   • When (x = 1), (y = \frac{1}{2}e^1 + 1 = \frac{1}{2}e + 1), so you can approximate the value of (y) or use a calculator to find the exact value.
   • Similarly, you can find other points or use technology to help plot more points accurately.

3. Determine the behavior:

   • As (x) approaches negative infinity, (e^x) approaches (0), so the graph approaches the horizontal asymptote (y = 1).
   • As (x) approaches positive infinity, (e^x) grows without bound, so the graph increases without bound.

4. Sketch the graph:

   • Start by plotting the key points and indicating the behavior as described above.
   • Draw a smooth curve that passes through the plotted points, reflecting the increasing behavior of (e^x) and the vertical stretch and shift.

By following these steps, you can accurately graph the function (y = \frac{1}{2}e^x + 1).