Lesson 4
Reflecting Functions
 Let’s reflect some graphs.
4.1: Notice and Wonder: Reflections
What do you notice? What do you wonder?
4.2: Reflecting Across
Here is the graph of function \(f\) and a table of values.
\(x\)  \(f(x)\)  \(g(x) = \textf(x)\) 

3  0  
1.5  4.3  
1  4  
0  1.8  
0.6  0  
2.6  3.9  
4  0 
 Let \(g\) be the function defined by \(g(x) = \textf(x)\). Complete the table.
 Sketch the graph of \(g\) on the same axes as the graph of \(f\) but in a different color.
 Describe how to transform the graph of \(f\) into the graph of \(g\). Explain how the equation produces this transformation.
4.3: Reflecting Across a Different Way
Here is another copy of the graph of \(f\) from the earlier activity. This time, let \(h\) be the function defined by \(h(x) = f(\textx)\).
 Use the definition of \(h\) to find \(h(0)\). Does your answer agree with your prediction?
 What does your prediction tell you about \(h(\text0.6)\)? Does your answer agree with the definition of \(h\)?

Complete the tables. The values for \(x\) will not be the same for the two tables.
\(x\) \(f(x)\) 3 0 1.5 4.3 1 4 0 1.8 0.6 0 2.6 3.9 4 0 \(x\) \(h(x)=f(\textx)\)  Sketch the graph of \(h\) on the same axes as the graph of \(f\) but in a different color.
 Describe what happened to the graph of \(f\) to transform it into the graph of \(h\). Explain how the equation produces this transformation.
 Describe how the graph of \(h\) relates to the graph of \(g\) defined in the earlier activity.
 Write an equation relating \(h\) and \(g\).
Summary
Here are graphs of the functions \(f\), \(g\), and \(h\), where \(g(x)=\textf(x)\) and \(h(x)=f(\textx)\). How do these equations match the transformation we see from \(f\) to \(g\) and from \(f\) to \(h\)?
Considering first the equation \(g(x)=\textf(x)\), we know that for the same input \(x\), the value of \(g(x)\) will be the opposite of the value of \(f(x)\). For example, since \(f(0)=1\), we know that \(g(0)=\textf(0)=\text1\). We can see this relationship in the graphs where \(g\) is the reflection of \(f\) across the \(x\)axis.
Looking at \(h(x)=f(\textx)\), this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and 1 are opposites, so \(h(1)=f(\text1)\) (and \(h(\text1)=f(1)\) is also true!). We can see this relationship in the graphs where \(h\) is the reflection of \(f\) across the \(y\)axis.