Reflection of Graphs of Functions

The reflections of graphs in the x-axis and y-axis are explored by comparing the graphs of \( f(x) \), \( -f(x) \), \( f(-x) \), and \( -f(-x) \):

The function \( f(x) \) is a quadratic function of the form:

\[ f(x) = a x^2 + b x + c \]

Adjust the parameters \( a \), \( b \), and \( c \) below to instantly visualize how function transformations and reflections behave on the coordinate plane.

\( a = \)
-10 +10

\( b = \)
-10 +10

\( c = \)
-10 +10

Tutorial & Guided Exploration

  1. Click on the "Draw" button above to render the initial graphs.
  2. Observe the four distinct graphs: \( f(x) \), \( -f(x) \), \( f(-x) \), and \( -f(-x) \). Test different values for \( a \), \( b \), and \( c \):
  3. Use what you have learned to explain the relationship between:
    • A. \( f(-x) \) and \( -f(-x) \)
    • B. \( f(x) \) and \( -f(-x) \)
    • C. \( -f(-x) \) and \( -f(x) \)

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