Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.

To graph the function , start by recognizing that the base indicates an exponential growth. The transformation from the parent function is a vertical shift upwards by 2 units.

The domain of is all real numbers , reflecting that you can input any real number for . The range, however, starts from 2 and goes to infinity since the minimum value occurs when approaches negative infinity—where approaches 0, making approach 2. The horizontal asymptote, therefore, is . Since the exponent in is positive, the function is increasing for all real numbers.

Now, here’s an eye-catching tidbit: Did you know that exponential functions like are frequently used to model real-world scenarios? They pop up in finance for compound interest calculations, in biology for population growth, and even in social media growth for user engagement patterns! Keep an eye out; they’re everywhere!

If you’re graphing this function, remember to watch for common pitfalls. One such mistake is forgetting to account for the vertical shift when plotting points. Also, while determining the asymptote, don’t mistakenly think it’s a point—it’s an invisible line guiding the behavior of the graph as it stretches horizontally! Keep these tips in mind to avoid confusion while graphing exponential functions!