Reflections of Graphs

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 = −x²is 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

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

REMEMBER that:

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

The graph of $y = \cos (x °)$ is shown on the graph below.

0VuoWNE2_ocr-7-graphs-transformations-reflections1

(a)

On the same graph sketch

y = - \cos (x °)

y = −cos(x) is a reflection in the x-axis
Reflect key points first- x-intercepts, maximums and minimums as shown below

(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

(b)

Comment on the graph of

y = \cos (- x °)

in relation to the graph of

y = \cos (x °)

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

The graph of $y = x^{2}$ 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

$y = - x^{2}$

And "translated 3 units left" leads to

$y = - {(x + 3)}^{2}$

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

Reflections of Graphs (OCR GCSE Maths)