Difference between revisions of "Single Transformations of Functions"

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==Introduction==
 
==Introduction==
[[File:Vertical shift.png|thumb|Vertical shift]]
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[[File:Vertical shift.png|thumb|Vertical shift: <math> f(x) = x^2 </math> (red) and <math> g(x) = x^2 + 4 </math> (blue)]]
[[File:Horizontal shift.png|thumb|Horizontal shift]]
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[[File:Horizontal shift.png|thumb|Horizontal shift: <math> f(x) = sin(x) </math> (red) and <math> g(x) = sin(x + \frac{\pi}{2}) </math> (blue)]]
[[File:Vertical reflection.png|thumb|Vertical reflection]]
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[[File:Vertical reflection.png|thumb|Vertical reflection: <math> f(x) = \sqrt{x} </math> (red) and <math> g(x) = -\sqrt{x} </math> (blue)]]
[[File:Horizontal reflection.png|thumb|Horizontal reflection]]
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[[File:Horizontal reflection.png|thumb|Horizontal reflection: <math> f(x) = e^x </math> (red) and <math> g(x) = e^{-x} </math> (blue)]]
  
 
===Translations===
 
===Translations===

Revision as of 18:03, 15 September 2021

Introduction

Vertical shift: (red) and (blue)
Horizontal shift: (red) and (blue)
Vertical reflection: (red) and (blue)
Horizontal reflection: (red) and (blue)

Translations

One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function , the function is shifted vertically k units. For example, is the function shifted up by 4 units. is the function shifted down by 7.7 units.

Given a function f, a new function , where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, is the graph of shifted 3 units to the right. is the function shifted units to the left.

Reflections

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

Given a function , a new function is a horizontal reflection of the function , sometimes called a reflection about the y-axis. For example, is a horizontal reflection of the function .

Resources