Difference between revisions of "Multiple Transformations of Functions"
| Line 1: | Line 1: | ||
See [[Single Transformations of Functions]] for more information on translating, reflecting, compressing, and stretching functions. | See [[Single Transformations of Functions]] for more information on translating, reflecting, compressing, and stretching functions. | ||
==Combining Functions== | ==Combining Functions== | ||
| − | + | ||
| − | [[File:Two shifts.png|thumb|Vertical and horizontal shift: f(x) = x^2 + x (red) and g(x) = (x - 3)^2 + (x - 3) + 5 (blue; f(x) shifted 3 units right and 5 units up)]] | + | [[File:Two shifts.png|thumb|Vertical and horizontal shift: <math> f(x) = x^2 + x (red) </math> and <math> g(x) = (x - 3)^2 + (x - 3) + 5 </math> (blue; f(x) shifted 3 units right and 5 units up)]] |
If <math> f(x) </math> is some function, then <math> g(x) = f(x - h) + k </math> is the function <math> f(x) </math> shifted h units horizontally (to the right for h > 0 and to the left for h < 0) and k units vertically (up for k > 0 and down for k < 0). For example, <math> g(x) = (x - 3)^2 + (x - 3) + 5 </math> is the function <math> f(x) = x^2 + x </math> shifted 3 units to the right and 5 units up. <math> g(x) = \sqrt{x + 2} - 6 </math> is the function <math> f(x) = \sqrt{x} </math> shifted 2 units to the left and 6 units down. | If <math> f(x) </math> is some function, then <math> g(x) = f(x - h) + k </math> is the function <math> f(x) </math> shifted h units horizontally (to the right for h > 0 and to the left for h < 0) and k units vertically (up for k > 0 and down for k < 0). For example, <math> g(x) = (x - 3)^2 + (x - 3) + 5 </math> is the function <math> f(x) = x^2 + x </math> shifted 3 units to the right and 5 units up. <math> g(x) = \sqrt{x + 2} - 6 </math> is the function <math> f(x) = \sqrt{x} </math> shifted 2 units to the left and 6 units down. | ||
| + | |||
| + | The order that two transformations are applied can change the resulting function. A vertical transformation and a horizontal transformation can be applied in either order and will result in the same function. However, two vertical transformations or two horizontal transformations can result in differing functions depending on the order they are applied in. | ||
| + | |||
| + | Examples: | ||
| + | * Vertical and horizontal shift: Let <math> f(x) = x^3 </math>. A 3 unit horizontal shift THEN a 3 unit vertical shift of <math> f(x) </math> will result in the function <math> g(x) = (x - 3) + 3 </math>. If we apply the vertical shift first instead, we will still get the result <math> g(x) = (x - 3) + 3 </math>. | ||
| + | * | ||
| + | * | ||
| + | * | ||
| + | * | ||
| + | * | ||
==Resources== | ==Resources== | ||
* [https://online.math.uh.edu/Math1330-unpaid/ch1/s13/CombTransf/Combining_Transformations_Math1330_s13.pdf Combining Transformations], University of Houston | * [https://online.math.uh.edu/Math1330-unpaid/ch1/s13/CombTransf/Combining_Transformations_Math1330_s13.pdf Combining Transformations], University of Houston | ||
* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/sequences-of-transformations/ Sequences of Transformations], Lumen Learning | * [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/sequences-of-transformations/ Sequences of Transformations], Lumen Learning | ||
Revision as of 14:47, 16 September 2021
See Single Transformations of Functions for more information on translating, reflecting, compressing, and stretching functions.
Combining Functions
If is some function, then 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) } shifted h units horizontally (to the right for h > 0 and to the left for h < 0) and k units vertically (up for k > 0 and down for k < 0). 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) = (x - 3)^2 + (x - 3) + 5 } 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) = x^2 + x } shifted 3 units to the right and 5 units up. 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 + 2} - 6 } 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) = \sqrt{x} } shifted 2 units to the left and 6 units down.
The order that two transformations are applied can change the resulting function. A vertical transformation and a horizontal transformation can be applied in either order and will result in the same function. However, two vertical transformations or two horizontal transformations can result in differing functions depending on the order they are applied in.
Examples:
- Vertical and horizontal shift: Let 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 } . A 3 unit horizontal shift THEN a 3 unit vertical shift 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) } will result in 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 g(x) = (x - 3) + 3 } . If we apply the vertical shift first instead, we will still get the result 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) = (x - 3) + 3 } .
Resources
- Combining Transformations, University of Houston
- Sequences of Transformations, Lumen Learning
