Which of the following describes the function shown in the table below?
| x | y |
| --- | ------- |
| -4 | 16 |
| -1 | 2 |
| 2 | 0.25 |
| 4 | 0.0625 |
| 5 | 0.03125 |
A. exponential, there is a continual rate of growth
B. exponential, there is a continual rate of decay or decrease
C. quadratic, there is a second degree change in the [tex]$y$[/tex]-values
D. quadratic, there is a constant difference between consecutive [tex]$y$[/tex]-values
Answer (1)
Calculate the ratios between consecutive y values and observe they are not constant.
Calculate the first and second differences of y values and observe they are not constant.
Test the exponential decay option and find that y = ( 0.5 ) x fits all the given points.
Conclude that the function is exponential with a continual rate of decay or decrease: exponential, there is a continual rate of decay or decrease .
Explanation
Understanding the Problem We are given a table of x and y values and asked to determine the type of function that describes the relationship between them. The options are exponential growth, exponential decay, quadratic with a second-degree change in y -values, and quadratic with a constant difference between consecutive y -values.
Checking for Exponential Function First, let's examine the ratios between consecutive y values. If the ratio is constant, the function is exponential. The ratios are calculated as follows:
Ratio between y values for x = − 1 and x = − 4 : 16 2 = 0.125 Ratio between y values for x = 2 and x = − 1 : 2 0.25 = 0.125 Ratio between y values for x = 4 and x = 2 : 0.25 0.0625 = 0.25 Ratio between y values for x = 5 and x = 4 : 0.0625 0.03125 = 0.5
Since the ratios are not constant, we cannot conclude that it is a simple exponential function.
Checking for Quadratic Function Now, let's examine the differences between consecutive y values. If the first differences are constant, the function is linear. If the second differences are constant, the function is quadratic.
First differences: 2 − 16 = − 14 0.25 − 2 = − 1.75 0.0625 − 0.25 = − 0.1875 0.03125 − 0.0625 = − 0.03125
Since the first differences are not constant, the function is not linear.
Second differences: − 1.75 − ( − 14 ) = 12.25 − 0.1875 − ( − 1.75 ) = 1.5625 − 0.03125 − ( − 0.1875 ) = 0.15625
Since the second differences are not constant, the function is not a simple quadratic function.
Testing Exponential Decay Let's consider the possibility of an exponential function of the form y = a ⋅ b x . From the table, we can observe that as x increases, y decreases, suggesting exponential decay. We can try to find a value of b such that y = b x .
When x = 2 , y = 0.25 = ( 0.5 ) 2 . When x = 4 , y = 0.0625 = ( 0.5 ) 4 . When x = 5 , y = 0.03125 = ( 0.5 ) 5 . This suggests that y = ( 0.5 ) x .
Let's check if y = ( 0.5 ) x holds for x = − 4 and x = − 1 . When x = − 4 , y = ( 0.5 ) − 4 = 2 4 = 16 . When x = − 1 , y = ( 0.5 ) − 1 = 2 .
Conclusion Since y = ( 0.5 ) x holds for all given points, the function is exponential with a continual rate of decay or decrease.
Examples
Exponential decay is a mathematical concept that describes the decrease of a quantity over time. A common real-world example is the decay of radioactive isotopes. For instance, Carbon-14 dating uses the exponential decay of Carbon-14 to estimate the age of organic materials. Understanding exponential decay helps scientists determine the age of fossils and artifacts, providing valuable insights into Earth's history and human civilization. Another example is the depreciation of a car's value over time.
