Lesson 4

Reflecting Functions

  • Let’s reflect some graphs.

4.1: Notice and Wonder: Reflections

What do you notice? What do you wonder?

Piecewise graph, f. Starting at negative 3 comma negative 2, graph is positive linear, then horizontal, then negative linear, then positive linear, then horizontal. Ends at 5 comma 1.
Graph of function g on a coordinate plane.
Graph of function h on a coordinate plane.

 

4.2: Reflecting Across

Here is the graph of function \(f\) and a table of values.

A graph of function f on a coordinate plane.
\(x\) \(f(x)\) \(g(x) = \text-f(x)\)
-3 0  
-1.5 -4.3  
-1 -4  
0 -1.8  
0.6 0  
2.6 3.9  
4 0  
  1. Let \(g\) be the function defined by \(g(x) = \text-f(x)\). Complete the table.
  2. Sketch the graph of \(g\) on the same axes as the graph of \(f\) but in a different color.
  3. 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 \(h(x) = f(\text-x)\).

A graph of function f on a coordinate plane.
  1. Use the definition of \(h\) to find \(h(0)\). Does your answer agree with your prediction?
  2. What does your prediction tell you about \(h(\text-0.6)\)? Does your answer agree with the definition of \(h\)?

  3. 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
    \(x\) \(h(x)=f(\text-x)\)
               
       
       
       
       
       
       
  4. Sketch the graph of \(h\) on the same axes as the graph of \(f\) but in a different color.
  5. Describe what happened to the graph of \(f\) to transform it into the graph of \(h\). Explain how the equation produces this transformation.


  1. Describe how the graph of \(h\) relates to the graph of \(g\) defined in the earlier activity.
  2. Write an equation relating \(h\) and \(g\).

Summary

Here are graphs of the functions \(f\), \(g\), and \(h\), where \(g(x)=\text-f(x)\) and \(h(x)=f(\text-x)\). How do these equations match the transformation we see from \(f\) to \(g\) and from \(f\) to \(h\)?

\(f(x)\)

A graph of function f on a coordinate plane.

\(g(x)=\text-f(x)\)

A graph of function g on a coordinate plane.

\(h(x) = f(\text-x)\)

A graph of function h on a coordinate plane.

Considering first the equation \(g(x)=\text-f(x)\), 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 \(g(0)=\text-f(0)=\text-1\). We can see this relationship in the graphs where \(g\) is the reflection of \(f\) across the \(x\)-axis.

Looking at \(h(x)=f(\text-x)\), this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and -1 are opposites, so \(h(1)=f(\text-1)\) (and \(h(\text-1)=f(1)\) is also true!). We can see this relationship in the graphs where \(h\) is the reflection of \(f\) across the \(y\)-axis.