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GCSEMathsOCRRevision Notes7. Graphs of Equations & FunctionsTransformations of GraphsReflections of Graphs

Reflections of Graphs (OCR GCSE Maths)

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Daniel I

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Daniel I

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Reflections of Graphs

How do we reflect graphs of functions?

  • Reflections in the y-axis: given an original equation in x, the graph of that equation will be reflected in the y-axis by a minus sign inside a bracket next to x
    • For example, in relation to y = x2;
      • y = (−x)2 is a reflection in the y-axis
    • Another example, in relation to y = sin(x) where x is in degrees;
      • y = sin(−x) is a reflection in the y-axis
  • Reflections in the x-axis: given an original equation in x, the graph of that equation will be reflected in the x-axis by a minus sign outside the bracket
    • For example, in relation to y = x2;
      • y = −x2 is a reflection in the x-axis
        • (note that y = −x2 is the same as y = −(x2) )
    • Another example, in relation to y = sin(x) where x is in degrees;
      • y = −sin(x) is a reflection in the x-axis

How do we describe reflections of graphs?

  • Some questions give a transformed a transformed graph of an equation with an original graph of an equation and ask you to describe the transformation
  • To describe a reflection fully, you must include;
    1. the transformation: "reflection"
    2. the direction; in the x-axis or in the y-axis

  • Remember that

    • if the "−" is inside the bracket, it's a reflection in the y-axis
    • if the "−" is outside the bracket, it's a reflection in the x-axis

Examiner Tip

REMEMBER that:

  • "−" next to x: the graph reflects in the x direction (therefore it is a reflection in the y-axis!)

Worked example

The graph of y equals cos open parentheses x degree close parentheses is shown on the graph below.

0VuoWNE2_ocr-7-graphs-transformations-reflections1

(a)
On the same graph sketch y equals negative cos open parentheses x degree close parentheses.

y = −cos(x) is a reflection in the x-axis
Reflect key points first- x-intercepts, maximums and minimums as shown below
owBUL9Uh_ocr-7-graphs-transformations-reflections2
(Notice that points on the x-axis don't change during a reflection in the x-axis. They are invariant)
Now join your new points with a curved line. The new curve should go through the key points shown in the answer below
opoC1b2h_ocr-7-graphs-transformations-reflections3
(b)
Comment on the graph of y equals cos open parentheses negative x degree close parentheses in relation to the graph of y equals cos open parentheses x degree close parentheses.

y = cos(−x) is a reflection in the y-axis. If we reflect y=cos(x) in the y-axis it maps exactly onto itself

They are the same graph

 

Worked example

The graph of y equals x squared is reflected in the x-axis and translated 3 units to the left.
Write down an equation of the translated graph.


At GCSE, the order in which transformations are performed will not affect the answer (it can at A-level or IB)
"Reflected in the x-axis" leads to 

y equals negative x squared

And "translated 3 units left" leads to 

Error converting from MathML to accessible text.

Notice the link between this topic and completing the square/ turning point of quadratic graphs!

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    Daniel I

    Author: Daniel I

    Expertise: Maths

    Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.