Multiple Transformations of Functions - Revision history

Lila: /* Combining Functions */

2021-09-16T21:50:05Z

Combining Functions

Lila: /* Combining Functions */

2021-09-16T21:01:16Z

Combining Functions

Lila: /* Combining Functions */

2021-09-16T20:59:50Z

Combining Functions

Lila: /* Combining Functions */

2021-09-16T20:55:07Z

Combining Functions

Lila: /* Combining Functions */

2021-09-16T20:54:25Z

Combining Functions

Lila: /* Combining Functions */

2021-09-16T20:50:12Z

Combining Functions

Lila at 20:47, 16 September 2021

2021-09-16T20:47:22Z

Lila at 20:17, 16 September 2021

2021-09-16T20:17:58Z

Lila: Created page with "See Single Transformations of Functions ==Resources== * [https://online.math.uh.edu/Math1330-unpaid/ch1/s13/CombTransf/Combining_Transformations_Math1330_s13.pdf Combining..."

2021-09-16T19:43:53Z

Created page with "See Single Transformations of Functions ==Resources== * [https://online.math.uh.edu/Math1330-unpaid/ch1/s13/CombTransf/Combining_Transformations_Math1330_s13.pdf Combining..."

New page

See [[Single Transformations of Functions]]
==Resources==
* [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

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 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.
+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 different vertical transformations or two different horizontal transformations can result in differing functions depending on the order they are applied in.
 Let <math> f(x) = x^3 + x^2 </math> for the following examples:
 * Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + (-x)^2 + 3 = -x^3 + x^2 + 3</math>. If we apply the horizontal reflection first instead, we will still get the same result.
 * Vertical shift and vertical reflection: A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + x^2 + 3) = -x^3 - x^2 - 3</math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 - x^2 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
-* Horizontal reflection and vertical stretch: A horizontal reflection, then a vertical stretch of 2, will give us the function <math> g(x) = 2(-x)^3 = -2x^3 </math>. Switching the order of the transformations will result in the same function.
+* Horizontal shift and vertical compression: A 3 unit horizontal shift, then a vertical compression of 2, will result in the function <math> g(x) = 2((x-3)^3 + (x-3)^2) </math>. If we apply the vertical compression first, we get the same result.
-* Horizontal reflection and horizontal stretch: A horizontal reflection, then a vertical stretch of 2, will give us the function <math> g(x) = 2(-x)^3 = -2x^3 </math>. Switching the order of the transformations will result in the same function.
+* Horizontal shift and horizontal compression: A 3 unit horizontal shift, then a horizontal compression of 2, will result in the function <math> g(x) = (2x-3)^3 + (2x-3)^2 </math>. However, if we apply the horizontal compression first, we get the function <math> g(x) = (2(x-3))^3 + (2(x-3))^2 = (2x - 6)^3 + (2x - 6)^2 </math>, which is not the same as our previous result <math> g(x) </math>.
 ==Resources==
 * [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

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 Let <math> f(x) = x^3 + x^2 </math> for the following examples:
-* Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + (-x)^2 + 3 = -x^3 + x^2 + 3</math>. If we apply the horizontal reflection first instead, we will still get the result <math> g(x) = (-x)^3 + 3 </math>.
+* Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + (-x)^2 + 3 = -x^3 + x^2 + 3</math>. If we apply the horizontal reflection first instead, we will still get the same result.
-* Vertical shift and vertical reflection: A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + 3) </math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
+* Vertical shift and vertical reflection: A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + x^2 + 3) = -x^3 - x^2 - 3</math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 - x^2 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
 * Horizontal reflection and vertical stretch: A horizontal reflection, then a vertical stretch of 2, will give us the function <math> g(x) = 2(-x)^3 = -2x^3 </math>. Switching the order of the transformations will result in the same function.
 * Horizontal reflection and horizontal stretch: A horizontal reflection, then a vertical stretch of 2, will give us the function <math> g(x) = 2(-x)^3 = -2x^3 </math>. Switching the order of the transformations will result in the same function.

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 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.
-Let <math> f(x) = x^3 </math> for the following examples:
+Let <math> f(x) = x^3 + x^2 </math> for the following examples:
-* Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + 3 </math>. If we apply the horizontal reflection first instead, we will still get the result <math> g(x) = (-x)^3 + 3 </math>.
+* Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + (-x)^2 + 3 = -x^3 + x^2 + 3</math>. If we apply the horizontal reflection first instead, we will still get the result <math> g(x) = (-x)^3 + 3 </math>.
 * Vertical shift and vertical reflection: A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + 3) </math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
+* Horizontal reflection and vertical stretch: A horizontal reflection, then a vertical stretch of 2, will give us the function <math> g(x) = 2(-x)^3 = -2x^3 </math>. Switching the order of the transformations will result in the same function.
+* Horizontal reflection and horizontal stretch: A horizontal reflection, then a vertical stretch of 2, will give us the function <math> g(x) = 2(-x)^3 = -2x^3 </math>. Switching the order of the transformations will result in the same function.
+*
+*

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 Let <math> f(x) = x^3 </math> for the following examples:
 * Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + 3 </math>. If we apply the horizontal reflection first instead, we will still get the result <math> g(x) = (-x)^3 + 3 </math>.
-* Vertical shift and vertical reflection: Let <math> f(x) = x^3 </math>. A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + 3) </math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
+* Vertical shift and vertical reflection: A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + 3) </math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
+*
+*

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 Let <math> f(x) = x^3 </math> for the following examples:
-* Vertical shift and horizontal reflection: A 3 unit horizontal reflection 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>.
+* Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of <math> f(x) </math>, will result in the function <math> g(x) = (-x)^3 + 3 </math>. If we apply the horizontal reflection first instead, we will still get the result <math> g(x) = (-x)^3 + 3 </math>.
-* Vertical shift and vertical reflection: Let <math> f(x) = x^3 </math>. A 3 unit horizontal reflection THEN a 3 unit vertical shift of <math> f(x) </math>
+* Vertical shift and vertical reflection: Let <math> f(x) = x^3 </math>. A 3 unit vertical shift, THEN a vertical reflection of <math> f(x) </math>, will result in the function <math> g(x) = -(x^3 + 3) </math>. However, if we apply the vertical reflection before the vertical shift, we will get the function <math> h(x) = -x^3 + 3 </math>, which is not equal to our previous result <math> g(x) </math>.
+*
+*

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 See [[Single Transformations of Functions]] for more information on translating, reflecting, compressing, and stretching functions.
 ==Combining Functions==
-===Combining vertical and horizontal shifts===
-[[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.
 ==Resources==
 * [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