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IBMaths: AI HLDPRevision Notes2. Functions2.5 Transformations of Graphs2.5.2 Reflections of Graphs

Reflections of Graphs (DP IB Maths: AI HL)

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Reflections of Graphs

What are reflections of graphs?

  • When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
  • For a reflection:
    • the graph is flipped about one of the coordinate axes
      • Its orientation changes
    • the size of the graph remains unchanged
  • A particular reflection is specified by an axis of symmetry:
    • space y equals 0
      • This is the x-axis
    • space x equals 0
      • This is the y-axis

Reflections What Is

What effects do horizontal reflections have on the graphs and functions?

  • A horizontal reflection of the graph space y equals f left parenthesis x right parenthesis about the y-axis is represented by 
    • space y equals f left parenthesis negative x right parenthesis
  • The x-coordinates change
    • Their sign changes
  • The y-coordinates stay the same
  • The coordinates left parenthesis x comma space y right parenthesis become left parenthesis negative x comma space y right parenthesis
  • Horizontal asymptotes stay the same
  • Vertical asymptotes change
    • x equals k becomes x equals negative k

Reflections statement_horiz_Illustration

What effects do vertical reflections have on the graphs and functions?

  • A vertical reflection of the graph space y equals f left parenthesis x right parenthesis about the x-axis is represented by
    • space minus y equals f left parenthesis x right parenthesis
    • This is often rearranged to space y equals negative f left parenthesis x right parenthesis
  • The x-coordinates stay the same
  • The y-coordinates change
    • Their sign changes
  • The coordinates left parenthesis x comma space y right parenthesis become left parenthesis x comma space minus y right parenthesis
  • Horizontal asymptotes change
    • space y equals k becomes space y equals negative k
  • Vertical asymptotes stay the same

Reflections statement_vert_Illustration

Worked example

The diagram below shows the graph of space y equals f left parenthesis x right parenthesis.

we-image

a)
Sketch the graph of space y equals negative f left parenthesis x right parenthesis.

2-5-2-ib-aa-sl-reflect-graph-a-we-solution

b)
Sketch the graph of space y equals f left parenthesis negative x right parenthesis.

2-5-2-ib-aa-sl-reflect-graph-b-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.