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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Dan

Last updated

9 May 2023

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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:
- $y = 0$
  - This is the x-axis
- $x = 0$
  - 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 $y = f (x)$ about the y-axis is represented by
- $y = f (- x)$
The x-coordinates change
- Their sign changes
The y-coordinates stay the same
The coordinates $(x, y)$ become $(- x, y)$
Horizontal asymptotes stay the same
Vertical asymptotes change
- $x = k$ becomes $x = - k$

Reflections statement_horiz_Illustration

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

A vertical reflection of the graph $y = f (x)$ about the x-axis is represented by
- $- y = f (x)$
- This is often rearranged to $y = - f (x)$
The x-coordinates stay the same
The y-coordinates change
- Their sign changes
The coordinates $(x, y)$ become $(x, - y)$
Horizontal asymptotes change
- $y = k$ becomes $y = - k$
Vertical asymptotes stay the same

Worked example

The diagram below shows the graph of $y = f (x)$ .

we-image

Sketch the graph of

y = - f (x)

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

Sketch the graph of

y = f (- x)

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

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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.