Which of the following best describes how the $y$ values are increasing from one interval to the next?
A. by adding 3
B. by multiplying by 9
C. by multiplying by 3
What does 3 represent in this exponential equation?
A. the intersection with the x-axis
B. the base
C. the value of each exponent
Input-output table for the function $y=3^x$
| x | y |
|----:|-----:|
| -2 | 1 / 9 |
| -1 | 1 / 3 |
| 0 | 1 |
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 4 | 81 |
Answer (2)
The y values increase by multiplying by 3.
3 represents the base in the exponential equation.
The function is y = 3 x .
The answer is multiplying by 3 and the base.
Explanation
Understanding the Problem We are given a table of values for the function y = 3 x and asked to determine how the y values are increasing from one interval to the next, and what the number 3 represents in the equation.
Analyzing the Increase in y Values To determine how the y values are increasing, we can calculate the ratio between consecutive y values. If the ratio is constant, then the y values are increasing by multiplication. If the difference is constant, then the y values are increasing by addition.
Calculating the Ratio Let's calculate the ratio between consecutive y values:
1/9 1/3 = 3 1 ⋅ 1 9 = 3 1/3 1 = 1 ⋅ 1 3 = 3 1 3 = 3 3 9 = 3 9 27 = 3 27 81 = 3
Since the ratio between consecutive y values is constant and equal to 3, the y values are increasing by multiplying by 3.
Identifying the Base In the exponential equation y = 3 x , the number 3 is the base. The base is the value that is raised to the power of x .
Conclusion Therefore, the y values are increasing by multiplying by 3, and 3 represents the base in the exponential equation.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, the population can be modeled by the exponential function P ( t ) = P 0 ⋅ 2 t , where P 0 is the initial population and t is the time in hours. Understanding exponential functions helps us predict and analyze these phenomena.
The y values in the function y = 3 x increase by multiplying by 3 at each interval. The number 3 in the equation represents the base of the exponential function. Therefore, the correct choices are 'C. by multiplying by 3' and 'B. the base.'
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