Final answer:
The function y=1/2*2ˣ⁺¹+1 has a horizontal shift of 1 unit left, a vertical compression by a factor of 1/2, and a vertical shift upward by 1 unit.
Step-by-step explanation:
To identify the transformations that have occurred in the equation y=\frac{1}{2}\cdot2^{x+1}+1, we break it down step by step:
- The base function is y=2^x, which represents an exponential function.
- To get from y=2^x to y=2^{x+1}, we perform a horizontal shift of 1 unit to the left, because adding inside the argument of the function (x+1) moves the graph in the opposite direction of the sign.
- The \frac{1}{2} multiplier in front of the function is a vertical compression by a factor of \( \frac{1}{2} \), making the graph more compact along the y-axis.
- Lastly, the +1 at the end of the function represents a vertical shift upward by 1 unit.
Summarizing, the equation y=\frac{1}{2}\cdot2^{x+1}+1 has undergone a horizontal shift left by 1 unit, a vertical compression by a factor of \( \frac{1}{2} \), and a vertical shift upward by 1 unit.