which graph represents a reflection of $f(x) = \\frac{1}{10}(10)^x$ across the $y$-axis?
which graph represents a reflection of $f(x) = \\frac{1}{10}(10)^x$ across the $y$-axis?
Answer
Explanation:
Step1: Recall Reflection Rule
To reflect a function ( f(x) ) across the ( y )-axis, we use the transformation ( f(-x) ). So for ( f(x)=\frac{1}{10}(10)^{x} ), the reflected function is ( g(x)=f(-x)=\frac{1}{10}(10)^{-x} ).
Step2: Simplify the Reflected Function
Simplify ( g(x)=\frac{1}{10}(10)^{-x} ). Using the exponent rule ( a^{-n}=\frac{1}{a^{n}} ), we get ( g(x)=\frac{1}{10}\left(\frac{1}{10}\right)^{x}=\frac{1}{10}(0.1)^{x} ). This is a decreasing exponential function (since the base ( 0.1<1 )) that passes through ( (0, \frac{1}{10}) ) (when ( x = 0 ), ( g(0)=\frac{1}{10}(0.1)^{0}=\frac{1}{10} )).
Step3: Analyze Original and Reflected Graphs
The original function ( f(x)=\frac{1}{10}(10)^{x} ) is an increasing exponential function (base ( 10>1 )) passing through ( (0, \frac{1}{10}) ). The reflected function ( g(x) ) is decreasing, so we look for a decreasing exponential graph that passes through ( (0, \frac{1}{10}) ).
Looking at the graphs:
- The first graph: Check if it's decreasing and passes through ( (0, \frac{1}{10}) ). It appears to be a decreasing exponential starting from the top left, passing near ( (0, \frac{1}{10}) ), which matches ( g(x) ).
- The second graph: Increasing exponential (matches original ( f(x) ), not the reflection).
- The third graph: Decreasing but let's check the ( y )-intercept. If ( x = 0 ), ( g(0)=\frac{1}{10} ), but this graph seems to have a different ( y )-intercept (maybe more negative or different scale? Unlikely).
- The fourth graph: Increasing? No, wait, no—wait, the fourth graph: Wait, no, let's re - check. Wait, the first graph: Let's confirm the ( y )-intercept. At ( x = 0 ), ( y=\frac{1}{10}=0.1 ), so the graph should cross the ( y )-axis at ( (0, 0.1) ). The first graph has a ( y )-intercept around ( 0.1 ) and is decreasing, which matches ( g(x)=\frac{1}{10}(0.1)^{x} ).
Answer:
The first graph (the left - most graph)