TRANSFORMATION OF FUNCTIONS: REFLECTION

A reflection is a transformation that flips a graph over a line, called the line of reflection.

Reflection about the x-axis and y-axis The graph of y = - f(x) is the reflection of the graph of y=f(x) about the x-axis.          The graph of y = f(-x) is the reflection of the graph of y=f(x) about the y-axis.

Examples

Example 1

Use the graph of f(x)= x2 to obtain the graph of g(x)= - x2.

Solution :

Since g(x)= - f(x) then the graph of g(x) is the reflection of the graph of f(x) about the x-axis.

First, graph f(x)= x2 (blue graph) and then reflect (flip) it about the x-axis to obtain the graph of

g(x)= - x2 (red graph).

Example 2

Try these problems

QUESTIONS

Use the graph of f(x) = |x| to obtain the graph of g(x)= -|x|.

Use the graph of f(x)=x3 to obtain the graph of g(x)= - x3.

Use the graph of f(x)=√x to obtain the graph of g(x)= - √x.

Use the graph of f(x)=√x to obtain the graph of g(x)= - √x..

ANSWERS

1. Use the graph of f(x) = |x| to obtain the graph of g(x)= -|x|.

2. Use the graph of f(x)=x3 to obtain the graph of g(x)= - x3.

3. Use the graph of f(x)=√x to obtain the graph of g(x)= - √x.

4. Use the graph of f(x)=√x to obtain the graph of g(x)= - √x..

1. To graph g(x)= -|x|, we first graph f(x) = |x| (blue graph) and then reflect it about the x-axis.

   The graph of g(x)= -|x| is the red graph.

   




2. To graph g(x)= - x3, we first graph f(x)=x3 (blue graph) and then reflect it about the x-axis.

   The graph of g(x)= - x3 is the red graph.

   




3. To graph g(x)= - √x, we first graph f(x)= √x (blue graph) and then reflect it about the x-axis.

   The graph of g(x)= - √x. is the red graph.

   Note that the domain of both functions is x ³ 0.

   




4. Use the graph of f(x)= √x to obtain the graph of g(x)= √-x.

   To graph g(x)= √-x, we first graph f(x)= √x (first graph) and then reflect it

   about the y-axis (second graph).

   Note that the domain of f(x) is x ³ 0 while that of g(x) is x £ 0.