<< Chapter < Page Chapter >> Page >

Reflecting a graph horizontally and vertically

Reflect the graph of s ( t ) = t (a) vertically and (b) horizontally.

  1. Reflecting the graph vertically means that each output value will be reflected over the horizontal t- axis as shown in [link] .

    Graph of the vertical reflection of the square root function.
    Vertical reflection of the square root function

    Because each output value is the opposite of the original output value, we can write

    V ( t ) = s ( t )  or  V ( t ) = t

    Notice that this is an outside change, or vertical shift, that affects the output s ( t ) values, so the negative sign belongs outside of the function.

  2. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in [link] .

    Graph of the horizontal reflection of the square root function.
    Horizontal reflection of the square root function

    Because each input value is the opposite of the original input value, we can write

    H ( t ) = s ( t )  or  H ( t ) = t

    Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.

    Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [ 0 , ) and range [ 0 , ) , the vertical reflection gives the V ( t ) function the range ( , 0 ] and the horizontal reflection gives the H ( t ) function the domain ( , 0 ] .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Reflect the graph of f ( x ) = | x 1 | (a) vertically and (b) horizontally.

  1. Graph of a vertically reflected absolute function.
  2. Graph of an absolute function translated one unit left.
Got questions? Get instant answers now!

Reflecting a tabular function horizontally and vertically

A function f ( x ) is given as [link] . Create a table for the functions below.

  1. g ( x ) = f ( x )
  2. h ( x ) = f ( x )
x 2 4 6 8
f ( x ) 1 3 7 11
  1. For g ( x ) , the negative sign outside the function indicates a vertical reflection, so the x -values stay the same and each output value will be the opposite of the original output value. See [link] .

    x 2 4 6 8
    g ( x ) –1 –3 –7 –11
  2. For h ( x ) , the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the h ( x ) values stay the same as the f ( x ) values. See [link] .

    x −2 −4 −6 −8
    h ( x ) 1 3 7 11
Got questions? Get instant answers now!
Got questions? Get instant answers now!

A function f ( x ) is given as [link] . Create a table for the functions below.

  1. g ( x ) = f ( x )
  2. h ( x ) = f ( x )
x −2 0 2 4
f ( x ) 5 10 15 20
  1. g ( x ) = f ( x )

    x -2 0 2 4
    g ( x ) 5 10 15 20
  2. h ( x ) = f ( x )

    x -2 0 2 4
    h ( x ) 15 10 5 unknown
Got questions? Get instant answers now!

Applying a learning model equation

A common model for learning has an equation similar to k ( t ) = 2 t + 1 , where k is the percentage of mastery that can be achieved after t practice sessions. This is a transformation of the function f ( t ) = 2 t shown in [link] . Sketch a graph of k ( t ) .

Graph of k(t)

This equation combines three transformations into one equation.

  • A horizontal reflection: f ( t ) = 2 t
  • A vertical reflection: f ( t ) = 2 t
  • A vertical shift: f ( t ) + 1 = 2 t + 1

We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).

  1. First, we apply a horizontal reflection: (0, 1) (–1, 2).
  2. Then, we apply a vertical reflection: (0, −1) (-1, –2).
  3. Finally, we apply a vertical shift: (0, 0) (-1, -1).

This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.

In [link] , the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.

Graphs of all the transformations.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask