Pre-Calculus 12 Notes

By: Gen L.

In partnership with Hyperion University, 2024

Lesson 2: Reflections

What are Reflections?

  • A transformation where points on the original graph have an image resulting in a reflection along a line.
  • Shape doesn't change, but orientation might.

Vertical Reflections

  • Defined as:

or

  • x-coordinate is identical ()
  • y-coordinate is inverted ()
  • Distance from the x-axis is identical, just on the other side.
  • Mapping Notation:

Example: Vertical Reflection

Given a function , graph .

Mapping Points:
A
B
C
D
E

On Invariant Points

  • Points that remain unchanged after a transformation.
    • In this case, any point on the x-axis.
  • Example: in the previous example,

Horizontal Reflections

  • Defined as:

  • x-coordinate is inverted ()
  • y-coordinate is identical ()
  • Distance from the y-axis is identical, just on the other side.
  • Mapping Notation:

Example: Horizontal Reflection

Given a function , graph .

Mapping Points:
A
B
C
D
E

Invariant Points

  • For this example: All points on the y-axis are invariant.
  • The only invariant point in the previous function is the y-intercept

Practice:

What would the following functions look like if they were reflected in the:

  • y-axis?
  • x-axis?

Answers:

  • y-axis:
  • x-axis:

Note:

  • Now, we can translate and reflect graphs.
  • It gets more complicated from here.

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