Transformation of Functions: Learn It 5

Identifying Horizontal Shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift.

horizontal shift

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

This shifts the graph of the function horizontally.

  • Rightward shift: If you subtract a constant [latex]c[/latex] from [latex]x[/latex] before applying the function [latex]f[/latex], the graph of the function shifts to the right by [latex]c[/latex] units.

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

  • Leftward shift: If you add a constant [latex]c[/latex] to [latex]x[/latex] before applying the function [latex]f[/latex], the graph of the function shifts to the left by [latex]c[/latex] units.

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

 

The image shows the graph of the cube root function [latex]f(x) = \sqrt[3]{x}[/latex] (solid blue line) and its horizontally shifted version [latex]f(x + 1)[/latex] (dashed orange line).Graph of f of x equals the cubed root of x shifted left one unit, the resulting graph passes through the point (0,-1) instead of (0,0), (0, 1) instead of (1,1) and (-2, -1) 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].

Horizontally 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 to the left of the corresponding point on the graph of [latex]f(x)[/latex].
  • For example:
    • If [latex]x=0[/latex], then [latex]\sqrt[3]{0+1} =  \sqrt[3]{1} = 1[/latex].
    • If [latex]x=-2[/latex], then [latex]\sqrt[3]{-2+1} = \sqrt[3]{-1} = -1[/latex].

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

The function [latex]G\left(m\right)[/latex] gives the number of gallons of gas required to drive [latex]m[/latex] miles. Interpret [latex]G\left(m\right)+10[/latex] and [latex]G\left(m+10\right)[/latex].

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

  1. Identify the input row or column.
  2. Determine the magnitude of the shift.
  3. Add the shift to the value in each input cell.

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 - 3\right)[/latex].

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