Difference between revisions of "Single Transformations of Functions"
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==Introduction== | ==Introduction== | ||
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| − | + | <div class="floatright"> | |
| − | + | <gallery perrow="2" widths="300px" heights="200px"> | |
| − | + | File:Vertical shift.png|Vertical shift: <math> f(x) = x^2 </math> (red) and <math> g(x) = x^2 + 4 </math> (blue) | |
| + | 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:Vertical reflection.png|Vertical reflection: <math> f(x) = \sqrt{x} </math> (red) and <math> g(x) = -\sqrt{x} </math> (blue) | ||
| + | File:Horizontal reflection.png|Horizontal reflection: <math> f(x) = e^x </math> (red) and <math> g(x) = e^{-x} </math> (blue) | ||
| + | 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) | ||
| + | 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) | ||
| + | </gallery> | ||
| + | </div> | ||
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===Translations=== | ===Translations=== | ||
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If a function satisfies neither of these conditions, then it is neither even nor odd. For example, <math> f(x) = x^2 + x </math> is neither even nor odd because <math> f(-x) = (-x)^2 + (-x) = x^2 - x </math>, which is not equal to <math> f(x) </math>, and <math> -f(-x) = -(-x)^2 - (-x) = -x^2 + x </math>, which is also not equal to <math> f(x) </math>. | If a function satisfies neither of these conditions, then it is neither even nor odd. For example, <math> f(x) = x^2 + x </math> is neither even nor odd because <math> f(-x) = (-x)^2 + (-x) = x^2 - x </math>, which is not equal to <math> f(x) </math>, and <math> -f(-x) = -(-x)^2 - (-x) = -x^2 + x </math>, which is also not equal to <math> f(x) </math>. | ||
| + | |||
| + | ===Compressions and Stretches=== | ||
| + | Given a function <math> f(x) </math>, a new function <math> g(x) = af(x) </math>, where <math> a </math> is a constant, is a vertical stretch or vertical compression of the function <math> f(x) </math>. | ||
| + | * If <math> a > 1 </math>, then the graph will be stretched. | ||
| + | * If <math> 0 < a < 1 </math>, then the graph will be compressed. | ||
| + | * If a < 0, then there will be a vertical stretch or compression of a factor of <math> |a| </math>, along with a vertical reflection. | ||
| + | |||
| + | Given a function <math> f(x) </math>, a new function <math> g(x) = f(bx) </math>, where <math> b </math> is a constant, is a horizontal stretch or horizontal compression of the function <math> f(x) </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 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 | ||
| − | * | + | |
| + | == Licensing == | ||
| + | Content obtained and/or adapted from: | ||
| + | * [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
Contents
Introduction
Vertical shift: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2 } (red) and (blue)
Horizontal shift: (red) and (blue)
Vertical reflection: (red) and (blue)
Horizontal reflection: (red) and (blue)
Vertical stretch/compression: (red), (blue, vert. compression), (green, vert. stretch), and (black, vert. stretch and vert. reflection)
Horizontal stretch/compression: (red), (blue, horiz. compression), (green, horiz. stretch), and (black, horiz. compression and horiz. reflection)
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2 } shifted 3 units to the right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = sin(x + \frac{\pi}{2}) } is the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = sin(x) } shifted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{2} } units to the left.
Reflections
Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , a new function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = -f(x) } is a vertical reflection of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , sometimes called a reflection about (or over, or through) the x-axis. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = -\sqrt{x} } is a vertical reflection of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \sqrt{x}} .
Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , a new function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = f(-x) } is a horizontal reflection of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , sometimes called a reflection about the y-axis. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = e^{-x} } is a horizontal reflection of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = e^x } .
Even and Odd Functions
A function f is even if for all values of x, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = f(-x) } ; that is, a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } is even if its horizontal reflection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(-x) } is identical to itself. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2 } is an even function since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(-x) = (-x)^2 = (-1)^2(x)^2 = x^2 = f(x)} .
A function f is odd if for all values of x, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = -f(-x) } ; that is, a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } is odd if a horizontal reflection and vertical reflection of the function results in the same function. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^3 } is an odd function since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -f(-x) = -(-x)^3 = (-1)(-1)^3(x)^3 = x^3 = f(x)} .
If a function satisfies neither of these conditions, then it is neither even nor odd. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2 + x } is neither even nor odd because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(-x) = (-x)^2 + (-x) = x^2 - x } , which is not equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , and , which is also not equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } .
Compressions and Stretches
Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , a new function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = af(x) } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } is a constant, is a vertical stretch or vertical compression of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } .
- If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 1 } , then the graph will be stretched.
- If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < a < 1 } , then the graph will be compressed.
- If a < 0, then there will be a vertical stretch or compression of a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a| } , along with a vertical reflection.
Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } , a new function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = f(bx) } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } is a constant, is a horizontal stretch or horizontal compression of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } .
- If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b > 1 } , then the graph will be horizontally compressed by a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{b} } .
- If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < b < 1 } , then the graph will be horizontally stretched by a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{b} } .
- If b < 0, then there will be a horizontal stretch or compression of a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|\frac{1}{b} \Big| } , along with a vertical reflection.
Resources
- Intro to Transformations of Functions, Lumen Learning
Licensing
Content obtained and/or adapted from:
- Transformations of Functions, Lumen Learning College Algebra under a CC BY-SA license
