Complete the table to investigate dilations of exponential functions.
| x | 2^x | 3 \cdot 2^x | 2^{3x} |
| --- | --- | ----------- | ------ |
| -2 | 1/4 | 3/4 | 1/64 |
| -1 | 1/2 | 3/2 | 1/8 |
| 0 | a | b | c |
| 1 | d | e | f |
| 2 | 4 | 12 | 64 |
Which function represents a vertical stretch of an exponential function?
y=2^x
y=3 \cdot 2^x
y=2^{3x}
Answer (2)
The completed table includes values calculated for the exponential functions, specifically y = 2 x , y = 3 ⋅ 2 x , and y = 2 3 x . The function that represents a vertical stretch of the exponential function is y = 3 ⋅ 2 x . Thus, the chosen option is y = 3 ⋅ 2 x .
;
Calculate the values for b , d , e , and f by substituting the given x values into the respective functions.
Complete the table with the calculated values.
Identify the function that represents a vertical stretch of y = 2 x , which is of the form y = k ⋅ 2 x .
Conclude that the function representing a vertical stretch is y = 3 ⋅ 2 x .
Explanation
Understanding the Problem We are given a table with values of x and corresponding values of 2 x , 3 ⋅ 2 x , and 2 3 x . We need to complete the table and identify which function represents a vertical stretch of y = 2 x .
Completing the Table First, let's complete the table. We are given that a = 1 and c = 1 . We need to find the values of b , d , e , and f .
Calculating b To find b , we substitute x = 0 into 3 ⋅ 2 x . Thus, b = 3 ⋅ 2 0 = 3 ⋅ 1 = 3 .
Calculating d To find d , we substitute x = 1 into 2 x . Thus, d = 2 1 = 2 .
Calculating e To find e , we substitute x = 1 into 3 ⋅ 2 x . Thus, e = 3 ⋅ 2 1 = 3 ⋅ 2 = 6 .
Calculating f To find f , we substitute x = 1 into 2 3 x . Thus, f = 2 3 ( 1 ) = 2 3 = 8 .
Completed Table Now we have the completed table:
x
2 x
3 ⋅ 2 x
2 3 x
-2
4 1
4 3
64 1
-1
2 1
2 3
8 1
0
1
3
1
1
2
6
8
2
4
12
64
Identifying Vertical Stretch A vertical stretch of y = 2 x is of the form y = k ⋅ 2 x where k is a constant. Comparing the given functions, y = 3 ⋅ 2 x is a vertical stretch of y = 2 x with k = 3 . The function y = 2 3 x can be rewritten as y = ( 2 3 ) x = 8 x , which is a horizontal compression, not a vertical stretch.
Final Answer Therefore, the function that represents a vertical stretch of an exponential function is y = 3 ⋅ 2 x .
Examples
Exponential functions are used to model population growth. A vertical stretch of an exponential function can represent a scenario where the initial population is larger. For example, if we model the growth of bacteria with 2 x , then 3 ⋅ 2 x would represent the growth of bacteria with three times the initial population. Understanding these transformations allows us to accurately model and predict real-world phenomena.
