To sketch the graph of the function [tex]\( f(x) = 2^{x+1} \)[/tex], we can understand how it is derived from the basic exponential function. Let’s break down the process:
1. Identify the Basic Exponential Function: The basic exponential function to compare with is [tex]\( y = 2^x \)[/tex].
2. Transform the Basic Function: The given function [tex]\( f(x) = 2^{x + 1} \)[/tex] involves a transformation from the base function. Specifically:
- The [tex]\( +1 \)[/tex] inside the exponent [tex]\( x + 1 \)[/tex] indicates a horizontal shift of the graph of [tex]\( y = 2^x \)[/tex].
3. Determine the Direction of the Shift: For exponential functions, adding a positive number inside the exponent ([tex]\( x + 1 \)[/tex]) results in a leftward shift of the graph. The expression [tex]\( x + 1 \)[/tex] shifts the graph of [tex]\( y = 2^x \)[/tex] one unit to the left.
Here’s a step-by-step description to obtain the graph:
- Start with the graph of [tex]\( y = 2^x \)[/tex]: This is the base exponential function.
- Shift the graph 1 unit to the left: Moving the graph horizontally to the left by 1 unit to get the graph of [tex]\( f(x) = 2^{x+1} \)[/tex].
Thus, the correct choice would be:
C. Start with the graph of [tex]\( y = 2^x \)[/tex]. Shift the graph 1 unit to the left.
Putting this in the answer box:
C. Start with the graph of [tex]\( y = 2^x \)[/tex]. Shift the graph 1 unit to the left.
