Difference between revisions of "Single Transformations of Functions"

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==Introduction==
 
==Introduction==
[[File:Vertical shift.png|thumb|Vertical shift: <math> f(x) = x^2 </math> (red) and <math> g(x) = x^2 + 4 </math> (blue)]]
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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)]]
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<div class="floatright">
[[File:Vertical reflection.png|thumb|Vertical reflection: <math> f(x) = \sqrt{x} </math> (red) and <math> g(x) = -\sqrt{x} </math> (blue)]]
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<gallery perrow="2" widths="300px" heights="200px">
[[File:Horizontal reflection.png|thumb|Horizontal reflection: <math> f(x) = e^x </math> (red) and <math> g(x) = e^{-x} </math> (blue)]]
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File:Vertical shift.png|Vertical shift: <math> f(x) = x^2 </math> (red) and <math> g(x) = x^2 + 4 </math> (blue)
[[File:Vertical stretch compression.png|thumb|Vertical stretch/compression: <math> f(x) = x^2 </math> (red), <math> g(x) = \frac{1}{2}x^2 </math> (blue, vert. compression), <math> h(x) = 2x^2 </math> (green, vert. stretch), and <math> g(x) = -2x^2 </math> (black, vert. stretch and vert. reflection)]]
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File:Horizontal shift.png|Horizontal shift: <math> f(x) = sin(x) </math> (red) and <math> g(x) = sin(x + \frac{\pi}{2}) </math> (blue)
[[File:Horizontal stretch compression.png|thumb|Horizontal stretch/compression: <math> f(x) = \sqrt(x) </math> (red), <math> g(x) = \sqrt(2x) </math> (blue, horiz. compression), <math> h(x) = \sqrt(\frac{1}{2}x) </math> (green, horiz. stretch), and <math> g(x) = -2x^2 </math> (black, horiz. compression and horiz. reflection)]]
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File:Vertical reflection.png|Vertical reflection: <math> f(x) = \sqrt{x} </math> (red) and <math> g(x) = -\sqrt{x} </math> (blue)
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File:Horizontal reflection.png|Horizontal reflection: <math> f(x) = e^x </math> (red) and <math> g(x) = e^{-x} </math> (blue)
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File:Vertical stretch compression.png|Vertical stretch/compression: <math> f(x) = x^2 </math> (red), <math> g(x) = \frac{1}{2}x^2 </math> (blue, vert. compression), <math> h(x) = 2x^2 </math> (green, vert. stretch), and <math> j(x) = -2x^2 </math> (black, vert. stretch and vert. reflection)
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File:Horizontal stretch compression.png|Horizontal stretch/compression: <math> f(x) = \sqrt{x} </math> (red), <math> g(x) = \sqrt{2x} </math> (blue, horiz. compression), <math> h(x) = \sqrt{\frac{1}{2}x} </math> (green, horiz. stretch), and <math> j(x) = \sqrt{-2x} </math> (black, horiz. compression and horiz. reflection)
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</gallery>
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</div>
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===Translations===
 
===Translations===
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* If <math> b > 1 </math>, then the graph will be horizontally compressed by a factor of <math> \frac{1}{b} </math>.
 
* If <math> b > 1 </math>, then the graph will be horizontally compressed by a factor of <math> \frac{1}{b} </math>.
 
* If <math> 0 < b < 1 </math>, then the graph will be horizontally stretched by a factor of <math> \frac{1}{b} </math>.
 
* If <math> 0 < b < 1 </math>, then the graph will be horizontally stretched by a factor of <math> \frac{1}{b} </math>.
* If b < 0, then there will be a horizontal stretch or compression of a factor of <math> |\frac{1}{b}| </math>, along with a vertical reflection.
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* If b < 0, then there will be a horizontal stretch or compression of a factor of <math> \Big|\frac{1}{b} \Big| </math>, along with a vertical reflection.
  
 
==Resources==
 
==Resources==
 
* [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/ Intro to Transformations of Functions], Lumen Learning
 
* [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/ Intro to Transformations of Functions], Lumen Learning
*
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== Licensing ==
 +
Content obtained and/or adapted from:
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* [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/ Transformations of Functions, Lumen Learning College Algebra] under a CC BY-SA license

Latest revision as of 12:39, 15 January 2022

Introduction


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 .

Even and Odd Functions

A function f is even if for all values of x, ; that is, a function is even if its horizontal reflection is identical to itself. For example, is an even function since .

A function f is odd if for all values of x, ; that is, a function is odd if a horizontal reflection and vertical reflection of the function results in the same function. For example, is an odd function since .

If a function satisfies neither of these conditions, then it is neither even nor odd. For example, is neither even nor odd because , which is not equal to , and , which is also not equal to .

Compressions and Stretches

Given a function , a new function , where is a constant, is a vertical stretch or vertical compression of the function .

  • If , then the graph will be stretched.
  • If , then the graph will be compressed.
  • If a < 0, then there will be a vertical stretch or compression of a factor of , along with a vertical reflection.

Given a function , a new function , where is a constant, is a horizontal stretch or horizontal compression of the function .

  • If , then the graph will be horizontally compressed by a factor of .
  • If , then the graph will be horizontally stretched by a factor of .
  • If b < 0, then there will be a horizontal stretch or compression of a factor of , along with a vertical reflection.

Resources

Licensing

Content obtained and/or adapted from: