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 = ?
Answer (2)
Calculate a = 2 0 = 1 .
Calculate b = 3 × 2 0 = 3 × 1 = 3 .
Calculate c = 2 3 × 0 = 2 0 = 1 .
Calculate d = 2 1 = 2 .
Calculate e = 3 × 2 1 = 3 × 2 = 6 .
The final answers are a = 1 , b = 3 , c = 1 , d = 2 , e = 6 . a = 1 , b = 3 , c = 1 , d = 2 , e = 6
Explanation
Understanding the Problem We are given a table with exponential functions and need to complete it by finding the values of a , b , c , d , and e . We will calculate each value using the given exponential expressions.
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 × 2 0 . Since 2 0 = 1 , we have b = 3 × 1 = 3 .
Calculating c Now, we need to find the value of c , which is 2 3 × 0 . Since 3 × 0 = 0 , we have c = 2 0 = 1 .
Calculating d Then, 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 Finally, we need to find the value of e , which is 3 × 2 1 . Since 2 1 = 2 , we have e = 3 × 2 = 6 .
Final Answer Therefore, the values are: a = 1 , b = 3 , c = 1 , d = 2 , and e = 6 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account with compound interest, the amount of money you have will grow exponentially over time. Similarly, the decay of a radioactive substance can be modeled using an exponential function. Understanding exponential functions helps us predict and analyze these phenomena.
The calculated values for the exponential functions are: a = 1, b = 3, c = 1, d = 2, and e = 6.
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