Question

Graphing Natural Exponential Functions In Exercise,sketch the graph of the function.See Example 1.

f(x) = e2x

Answer 1

The graph is shown below.

Explanation:

Given:

The exponential function to graph is given as:

`f(x)=e^(2x)`

In order to graph the above function, we first find some points on it by taking random values of 'x'.

x -2 -1 0 0.5

f(x) 0.018 0.135 1 2.718

Now, the points on the graph are:

(-2, 0.018), (-1, 0.135), (0, 1), and (0.5, 2.718).

Now, we plot these points on the graph.

Next, we find the horizontal asymptotes. For that, we find the limit with x tending to negative infinity. This gives,

`\lim_(x \to -\infty) f(x)\\\\ \lim_(x \to -\infty) e^(2x)\\\\  \lim_(x \to -\infty) e^(-\infty)=0`

Therefore,
`y=0` or the x-axis is the horizontal asymptote. As exponential functions are increasing functions, so for 'x' tending to positive infinity, the function will also tend towards positive infinity.

Now, we draw a smooth curve passing through the given points and continuing the graph parallel to x-axis for greater values of x along the negative x-axis.

The graph is shown below.