How do you graph #y=1/2e^x+1#?
Start with the graph for
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]}
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To graph the function (y = \frac{1}{2}e^x + 1), follow these steps:

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.

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.

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.

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).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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