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 |
a = ?
b = ?
c = ?
d = ?
e = ?
f = ?
Answer (2)
The completed values for the table are: a = 1, b = 3, c = 1, d = 2, e = 6, and f = 8. Each value corresponds to the respective exponential functions evaluated at specific x values. This demonstrates the properties of exponential growth and dilations effectively.
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Calculate a = 2 0 = 1 .
Calculate b = 3 c d o t 2 0 = 3 .
Calculate c = 2 3 c d o t 0 = 1 .
Calculate d = 2 1 = 2 , e = 3 c d o t 2 1 = 6 , f = 2 3 c d o t 1 = 8 .
The completed values are a = 1 , b = 3 , c = 1 , d = 2 , e = 6 , f = 8 , so the answer is a = 1 , b = 3 , c = 1 , d = 2 , e = 6 , f = 8 .
Explanation
Understanding the Problem We are given a table with values of three exponential functions: 2 x , 3 c d o t 2 x , and 2 3 x for x = − 2 , − 1 , 0 , 1 , 2 . Our task is to complete the table by finding the values of a , b , c , d , e , f .
Calculating a First, we need to find the value of a , which is 2 0 . Any number raised to the power of 0 is 1. Therefore, a = 2 0 = 1 .
Calculating b Next, we need to find the value of b , which is 3 c d o t 2 0 . Since 2 0 = 1 , we have b = 3 c d o t 1 = 3 .
Calculating c Now, we need to find the value of c , which is 2 3 c d o t 0 . Since 3 c d o t 0 = 0 , we have c = 2 0 = 1 .
Calculating d Next, we need to find the value of d , which is 2 1 . Any number raised to the power of 1 is the number itself. Therefore, d = 2 1 = 2 .
Calculating e Now, we need to find the value of e , which is 3 c d o t 2 1 . Since 2 1 = 2 , we have e = 3 c d o t 2 = 6 .
Calculating f Finally, we need to find the value of f , which is 2 3 c d o t 1 . Since 3 c d o t 1 = 3 , we have f = 2 3 = 2 c d o t 2 c d o t 2 = 8 .
Final Answer Therefore, we have found all the missing values: a = 1 , b = 3 , c = 1 , d = 2 , e = 6 , and f = 8 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. In finance, understanding exponential growth helps in calculating investment returns over time. For instance, if you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . This formula demonstrates how an initial investment grows exponentially over time, highlighting the power of compounding.
