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 (1)
Rewrite the second equation: y = ( 3 1 ) x = ( 3 − 1 ) x .
Simplify the second equation using the power of a power rule: y = ( 3 − 1 ) x = 3 − x .
Compare the two equations: y = 3 − x and y = 3 − x are the same.
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 , and we want to determine how their graphs compare. The options are that the graphs are the same, reflected across the x -axis, or reflected across the y -axis.
Rewriting the Second Equation Let's rewrite the second equation using the property that a 1 = a − 1 . So, we have y = ( 3 1 ) x = ( 3 − 1 ) x
Simplifying the Second Equation Using the power of a power rule, ( a m ) n = a mn , we can simplify the second equation further: y = ( 3 − 1 ) x = 3 − 1 ⋅ x = 3 − x
Comparing the Equations Now, we compare the two equations. The first equation is y = 3 − x , and the simplified second equation is also y = 3 − x . Since the equations are identical, their graphs are the same.
Conclusion Therefore, the graph of y = 3 − x is the same as the graph of y = ( 3 1 ) x .
Examples
Imagine you are designing a seesaw where the height on one side decreases exponentially as the height on the other side increases. The functions y = 3 − x and y = ( 3 1 ) x both describe this type of relationship. Understanding that these functions are equivalent helps you predict and balance the seesaw's motion accurately. This concept is also useful in modeling radioactive decay or the depreciation of an asset over time.
