Graph f(x)=2^x+1

Problem

ƒ(x)=2+1

Solution

1. Identify the parent function and the transformation. The parent function is g(x)=2 which is a basic exponential growth function.

2. Determine the vertical shift by looking at the constant term. The +1 indicates that the entire graph of 2 is shifted upward by 1 unit.

3. Find the horizontal asymptote of the function. Since the parent function has a horizontal asymptote at y=0 the shifted function ƒ(x)=2+1 has a horizontal asymptote at y=1

4. Calculate the y-intercept by evaluating the function at x=0

ƒ(0)=2+1

ƒ(0)=1+1=2

5. Plot additional points to determine the shape of the curve.
   For x=1

ƒ(1)=2+1=3

For x=2

ƒ(2)=2+1=5

For x=−1

ƒ*(−1)=2(−1)+1=1.5

6. Sketch the graph by drawing a smooth curve through the points (0,2) (1,3) (2,5) and (−1,1.5) ensuring the curve approaches the line y=1 as x decreases toward negative infinity.

Final Answer

The graph of ƒ(x)=2+1 is an exponential growth curve with a horizontal asymptote at y=1 and a y-intercept at (0,2)

Want more problems? Check here!