How does the graph of $y=3^{-x}$ compare to the graph of $y=\left(\frac{1}{3}\right)^x$?
A. The graphs are the same.
B. The graphs are reflected across the $x$-axis.
C. The graphs are reflected across the $y$-axis.
Answer (2)
The graphs of y = 3 − x and y = ( 3 1 ) x are the same because they represent identical expressions after rewriting. Hence, the correct answer is: The graphs are the same.
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Rewrite y = 3 − x as y = ( 3 − 1 ) x .
Simplify to get y = ( 3 1 ) x .
Compare the simplified equation with y = ( 3 1 ) x .
Conclude that the graphs are the same: The graphs are the same.
Explanation
Understanding the Problem We are given two functions: y = 3 − x and y = ( 3 1 ) x . We need to determine how their graphs compare.
Rewriting the First Equation Let's rewrite the first equation using the properties of exponents. Recall that a − b = a b 1 = ( a 1 ) b . Therefore, we can rewrite y = 3 − x as y = ( 3 − 1 ) x = ( 3 1 ) x .
Comparing the Equations Now we compare the rewritten equation y = ( 3 1 ) x with the second equation y = ( 3 1 ) x . We can see that they are exactly the same.
Conclusion Since the two equations are identical, their graphs are the same. Therefore, the graph of y = 3 − x is the same as the graph of y = ( 3 1 ) x .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding how different exponential functions relate to each other, such as comparing y = 3 − x and y = ( 1/3 ) x , helps in analyzing and predicting these phenomena. For example, in finance, understanding exponential decay can help in calculating the depreciation of an asset over time, while in biology, it can be used to model the decay of a drug in the bloodstream.
