Transformation of Functions: Learn It 4

Identifying Vertical Shifts

One simple kind of transformation involves shifting the entire graph of a function up and down, known as a vertical shift. 

vertical shift

A vertical shift occurs when you add or subtract a constant value to the function [latex]f(x)[/latex].

This shifts the graph of the function vertically without changing its shape.

  • Upward shift: If you add a constant [latex]c[/latex] to the function [latex]f(x)[/latex], the graph of the function shifts upward by [latex]c[/latex] units.

[latex]g(x) = f(x) +c[/latex]

  • Downward shift: If you subtract a constant [latex]c[/latex] to the function [latex]f(x)[/latex], the graph of the function shifts downward by [latex]c[/latex] units.

[latex]h(x) = f(x) -c[/latex]

The image below shows the graph of a function [latex]f(x)[/latex] (solid blue line) and its vertically shifted version [latex]f(x) + 1[/latex] (dashed orange line).Graph of f of x equals the cubed root of x shifted upward one unit, the resulting graph passes through the point (0,1) instead of (0,0), (1, 2) instead of (1,1) and (-1, 0) instead of (-1, -1)Original Function [latex]f(x)[/latex]

  • The solid blue curve represents the original function [latex]\sqrt[3]{x}[/latex].
  • The function [latex]f(x)[/latex] passes through the origin [latex](0,0)[/latex] because [latex]\sqrt[3]{0} = 0[/latex].

Vertically Shifted Function [latex]f(x) + 1[/latex]

  • The dashed orange curve represents the function  [latex]f(x) + 1  = \sqrt[3]{x} + 1[/latex].
  • Each point on the graph of [latex]f(x) + 1[/latex] is exactly [latex]1[/latex] unit higher than the corresponding point on the graph of [latex]f(x)[/latex].
  • For example:
    • If [latex]x=0[/latex], then [latex]\sqrt[3]{0} +1 = 0 + 1  = 1[/latex].
    • If [latex]x=1[/latex], then [latex]\sqrt[3]{1} +1 = 1 + 1  = 2[/latex].

A vertical shift involves moving the graph of a function up or down without altering its shape. In this case, adding [latex]1[/latex] to the function [latex]f(x) = \sqrt[3]{x}[/latex] results in a vertical shift of the graph upward by [latex]1[/latex] unit.

How To: Given a tabular function, create a new row to represent a vertical shift.

  1. Identify the output row or column.
  2. Determine the magnitude of the shift.
  3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.

A function [latex]f\left(x\right)[/latex] is given below. Create a table for the function [latex]g\left(x\right)=f\left(x\right)-3[/latex].

[latex]x[/latex] 2 4 6 8
[latex]f\left(x\right)[/latex] 1 3 7 11