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Wize Grade 11 Mathematics Textbook > Transformations on Functions

Reflections

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Reflections

Reflections

A reflection on a function is when it has been flipped over a vertical or horizontal line like either the x or y-axis.

Example 1



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Reflections from the equation

We can apply a reflection to a function by multiplying or dividing by -1. The graph will be flipped vertically or horizontally. Sometimes this is described as flipping over the x-axis or y-axis.
y=−f(x)y=f(−x)\begin{aligned} y&=\colorTwo{-}f(x) \\ y&=f(\colorTwo{-}x) \end{aligned}

Vertical Reflection (over x-axis)
  • Multiply the entire function by -1

Horizontal Reflection (over y-axis)
  • Multiplying the input variable by -1

Example 2
For each equation, describe the original function, and the type of reflection.

1. y=−2x2y = -2x^2
  • Function: y=x2y=x^2
  • Reflection: Vertical reflection over the x-axis.

2. y=(−x)3y=(-x)^3
  • Function: y=x3y =x^3
  • Reflection: Horizontal reflection over the y-axis

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Example: Reflections



In a printing press the image on an inked plate is a reflection of the final image on paper. This reflection can either be over the x-axis or y-axis depending on how the plate is used to stamp on the paper.

Suppose we have a plate that is horizontal reflection of our final paper over the y-axis.
We want to print the following graph of a function for the financial section.



1. What should the graph on the inked plate look like to produce the desired result?


2. Describe how the coordinates of key points on the graph change from the ink plate to the final paper. Use an example help describe the process.

All of the x-values change sign.
A key point on the ink plate like (−3,5)(-3, 5) becomes (3,5)(3, 5) on the final paper.

Practice: Reflections



Given that the original function is f(x)=xf(x) = \sqrt{x}
Fill in the blanks by describing over which axis the function will be reflected.
1. y=−xy = \sqrt{-x}
2. y=−xy = -\sqrt{x}

Practice: Reflections

The given table represents the inputs and outputs of a given function. If we reflect this function over the y-axis, what will be the new input and output values?

Complete the table with these new values transformed values.

xy
1
00
1
-2

Practice: Reflections

The graph of a function f(x)f(x) is given as well as its transformed graph.
Write the equation of the transformed graph.