Which of the following exponential equations could be represented by the table below?
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|--------|--------|--------|--------|--------|--------|---|---|----|
| y | 2.0014 | 2.0041 | 2.0123 | 2.037 | 2.111 | 2.333 | 3 | 5 | 11 |
A. [tex]$y=\left(\frac{1}{3}\right)^{x-3}-3$[/tex]
B. [tex]$y=3^{x-7}-3$[/tex]
C. [tex]$y=3^{x-3}+2$[/tex]
D. [tex]$y=\left(\frac{1}{3}\right)^{x-3}-2$[/tex]
Answer (1)
Substitute the given x values into each of the four equations.
Compare the calculated y values with the y values provided in the table.
Identify the equation that produces y values closest to those in the table.
Conclude that the equation represented by the table is y = 3 x − 3 + 2 .
Explanation
Problem Analysis We are given a table of x and y values and asked to identify which of the four given exponential equations could be represented by the table. The strategy is to substitute the x values from the table into each equation and see which equation produces y values that closely match the table's y values.
Testing Equation 1 Let's test the first equation: y = ( 3 1 ) x − 3 − 3 . We'll substitute the x values from the table into this equation.
Testing Equation 2 Let's test the second equation: y = 3 x − 7 − 3 . We'll substitute the x values from the table into this equation.
Testing Equation 3 Let's test the third equation: y = 3 x − 3 + 2 . We'll substitute the x values from the table into this equation.
Testing Equation 4 Let's test the fourth equation: y = ( 3 1 ) x − 3 − 2 . We'll substitute the x values from the table into this equation.
Comparison and Conclusion After substituting the x values into each of the four equations, we can compare the resulting y values with the values in the table. From the tool output, we can see that the third equation, y = 3 x − 3 + 2 , produces y values that closely match the table's y values.
Verification The y values for the equation y = 3 x − 3 + 2 are:
x
-3
-2
-1
0
1
2
3
4
5
y
2.00137
2.00412
2.01235
2.03704
2.11111
2.33333
3
5
11
These values closely match the given table.
Final Answer Therefore, the exponential equation that could be represented by the table is y = 3 x − 3 + 2 .
Examples
Exponential equations 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 size can be modeled by an exponential equation. Similarly, the amount of a radioactive substance remaining after a certain time can be modeled by an exponential decay equation. Understanding how to work with exponential equations allows us to make predictions and analyze these phenomena.
