To graph the function $f(x) = 2^x + 1$, follow these steps:

1. Identify the type of function: This is an exponential function where the base is 2, and the graph is shifted vertically by +1.

2. Key points to plot:

   • When $x = 0$, $f(x) = 2^0 + 1 = 1 + 1 = 2$.
   • When $x = 1$, $f(x) = 2^1 + 1 = 2 + 1 = 3$.
   • When $x = -1$, $f(x) = 2^{-1} + 1 = \frac{1}{2} + 1 = 1.5$.
   • When $x = 2$, $f(x) = 2^2 + 1 = 4 + 1 = 5$.
   • As $x \to -\infty$, $f(x) \to 1$, which represents the horizontal asymptote at $y = 1$.

3. General shape:

   • The graph is increasing for all $x$, as it's an exponential growth function.
   • The horizontal asymptote is at $y = 1$, meaning the function approaches 1 but never touches it as $x \to -\infty$.
   • The function crosses the point $(0, 2)$ and grows rapidly as $x \to \infty$.

4. Domain and range:

   • Domain: $(-\infty, \infty)$ (all real numbers)
   • Range: $(1, \infty)$ (the function value is always greater than 1)

Would you like to see a plotted graph or have any questions?

Here are 5 related questions to deepen your understanding:

1. What happens to the graph of $f(x) = 2^x + 1$ if we change the base from 2 to 3?
2. How does the vertical shift affect the graph of an exponential function?
3. What is the effect of a negative exponent on the value of an exponential function?
4. How would the graph change if we subtracted 1 instead of adding 1 in the function $f(x) = 2^x + 1$?
5. What are the characteristics of the horizontal asymptote for other exponential functions?

Tip: To better understand exponential functions, try plotting different values for the base and observe how they influence the growth rate and overall shape.