To sketch the graph of the function [tex]\( f(x) = 2^{x+1} \)[/tex], let's start by understanding how it relates to the basic exponential function [tex]\( y = 2^x \)[/tex].
### Step-by-Step Solution:
1. Basic Exponential Function:
- Begin with the graph of the basic exponential function [tex]\( y = 2^x \)[/tex]. This graph passes through the point (0,1) because [tex]\( 2^0 = 1 \)[/tex], and it approaches zero as [tex]\( x \)[/tex] approaches negative infinity. It grows rapidly as [tex]\( x \)[/tex] increases.
2. Understanding [tex]\( f(x) = 2^{x+1} \)[/tex]:
- The function given is [tex]\( f(x) = 2^{x+1} \)[/tex]. We need to rewrite this equation to see how it is transformed from the basic function [tex]\( y = 2^x \)[/tex].
3. Rewrite [tex]\( f(x) = 2^{x+1} \)[/tex]:
- Notice that [tex]\( f(x) = 2^{x+1} = 2^x \cdot 2^1 = 2 \cdot 2^x \)[/tex]. This tells us that the graph of [tex]\( f(x) \)[/tex] is essentially a transformation of the graph of [tex]\( y = 2^x \)[/tex].
4. Shift the Graph:
- [tex]\( f(x) = 2^{x+1} \)[/tex] can be interpreted as the graph of [tex]\( y = 2^x \)[/tex] shifted to the left by 1 unit. Why? The additional [tex]\( +1 \)[/tex] inside the exponent translates the graph horizontally.
### Transformations (Graphical Explanation):
- Horizontal Shift:
- The term [tex]\( x+1 \)[/tex] inside the exponent indicates a horizontal shift to the left by 1 unit. This is due to inverse operations inside the function. To see why, consider what happens to the input [tex]\( x \)[/tex] that would make the exponent zero:
- For [tex]\( y = 2^x \)[/tex], [tex]\( x = 0 \)[/tex] makes the exponent zero.
- For [tex]\( f(x) = 2^{x+1} \)[/tex], [tex]\( x = -1 \)[/tex] makes the exponent zero.
- Thus, every [tex]\( x \)[/tex] in [tex]\( y = 2^x \)[/tex] corresponds to [tex]\( x-1 \)[/tex] in [tex]\( f(x) = 2^{x+1} \)[/tex], effectively shifting the graph to the left by one unit.
### Graph Description and Comparison with Calculator:
- Use a graphing calculator to plot both [tex]\( y = 2^x \)[/tex] and [tex]\( f(x) = 2^{x+1} \)[/tex]. Verify that:
- The shape of the graph remains the same.
- Each point on [tex]\( y = 2^x \)[/tex] moves 1 unit to the left to form [tex]\( f(x) = 2^{x+1} \)[/tex].
Therefore, the correct transformation to describe [tex]\( f(x) = 2^{x+1} \)[/tex] is:
- Shift the graph of [tex]\( y = 2^x \)[/tex] to the left by 1 unit.
