Reflections of Graphs (Edexcel IGCSE Maths)

Spec point — 3.3B

Reflections of graphs

What are reflections of graphs?

• Reflections of graphs are a type of transformation where the curve is reflected about one of the axes

Examples of reflections

How do I reflect graphs?

• Let $y= \mathrm{f} (x)$ be the equation of the original graph

Vertical reflections: y=-f(x)

• $y=− \mathrm{f} (x)$ is a reflection in the $x$-axis

  • The $y$ coordinates change sign

    • The $x$ coordinates are unaffected

Example of a vertical reflection

Horizontal reflections: y=f(-x)

• $y= \mathrm{f} (−x)$ is a reflection in the $y$-axis

  • The $x$ coordinates change sign

    • The $y$ coordinates are unaffected

Example of a horizontal reflection

What happens to asymptotes when a graph is reflected?

• Any asymptotes of $\mathrm{f} (x)$ are also reflected

Exam Hint When reflecting graphs in the exam, reflect any key points on the graph first, then join them up with a smooth curve.

How does a reflection affect the equation of the graph?

• When a graph is reflected, you can change its equation algebraically

  • There is no need to sketch the graph

• Reflecting in the $x$-axis puts a $-$in front of the whole equation

  • For example, $y=x^{2}+2x$ becomes y=−(x²+2x)

    • This simplifies to $y=-x^{2}-2x$

• Reflecting in the $y$-axis replaces any$x$with(−x) in the equation

  • For example, $y=x^{2}+2x$ becomes y=(−x)²+2 (−x)

    • This simplifies to $y=x^{2}-2x$

How do I apply a combined reflection?

• The graph of $y=− \mathrm{f} (−x)$ is a combined reflection in both the $x$ and $y$ axes

  • It does not matter which order you apply these in

    • For example, reflect about the $y$-axis then about the $x$-axis

Worked Example The diagram below shows the graph of $y= \mathrm{f} (x)$.

Sketch the graph of $y= \mathrm{f} (−x)$.

Answer:

The transformation $y= \mathrm{f} (−x)$ is a reflection in the $y$-axis Reflect the points (-2, 6) and (1, -3) in the $y$-axis to get (2, 6) and (-1, -3) Sketch these points and join with a smooth curve through the origin