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Given a function and both a vertical and a horizontal shift, sketch the graph.

  1. Identify the vertical and horizontal shifts from the formula.
  2. The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.
  3. The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.
  4. Apply the shifts to the graph in either order.

Graphing combined vertical and horizontal shifts

Given f ( x ) = | x | , sketch a graph of h ( x ) = f ( x + 1 ) 3.

The function f is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of h has transformed f in two ways: f ( x + 1 ) is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in f ( x + 1 ) 3 is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in [link] .

Let us follow one point of the graph of f ( x ) = | x | .

  • The point ( 0 , 0 ) is transformed first by shifting left 1 unit: ( 0 , 0 ) ( −1 , 0 )
  • The point ( −1 , 0 ) is transformed next by shifting down 3 units: ( −1 , 0 ) ( −1 , −3 )
Graph of an absolute function, y=|x|, and how it was transformed to y=|x+1|-3.

[link] shows the graph of h .

The final function y=|x+1|-3.
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Given f ( x ) = | x | , sketch a graph of h ( x ) = f ( x 2 ) + 4.

Graph of h(x)=|x-2|+4.
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Identifying combined vertical and horizontal shifts

Write a formula for the graph shown in [link] , which is a transformation of the toolkit square root function.

Graph of a square root function transposed right one unit and up 2.

The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as

h ( x ) = f ( x 1 ) + 2

Using the formula for the square root function, we can write

h ( x ) = x 1 + 2
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Write a formula for a transformation of the toolkit reciprocal function f ( x ) = 1 x that shifts the function’s graph one unit to the right and one unit up.

g ( x ) = 1 x - 1 + 1

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Graphing functions using reflections about the axes

Another transformation that can be applied to a function is a reflection over the x - or y -axis. A vertical reflection reflects a graph vertically across the x -axis, while a horizontal reflection reflects a graph horizontally across the y -axis. The reflections are shown in [link] .

Graph of the vertical and horizontal reflection of a function.
Vertical and horizontal reflections of a function.

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x -axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y -axis.

Reflections

Given a function f ( x ), a new function g ( x ) = f ( x ) is a vertical reflection    of the function f ( x ) , sometimes called a reflection about (or over, or through) the x -axis.

Given a function f ( x ) , a new function g ( x ) = f ( x ) is a horizontal reflection    of the function f ( x ) , sometimes called a reflection about the y -axis.

Given a function, reflect the graph both vertically and horizontally.

  1. Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x -axis.
  2. Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the y -axis.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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