Difference between revisions of "Multiple Transformations of Functions"

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Let <math> f(x) = x^3 </math> for the following examples:
 
Let <math> f(x) = x^3 </math> for the following examples:
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* 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>.
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* 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>.
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* 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>
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* 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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Revision as of 14:54, 16 September 2021

See Single Transformations of Functions for more information on translating, reflecting, compressing, and stretching functions.

Combining Functions

Vertical and horizontal 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 + x (red) } and (blue; f(x) shifted 3 units right and 5 units up)

If is some function, then is the function 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, is the function 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.

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 } for the following examples:

  • Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection 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 horizontal reflection 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 } .
  • Vertical shift and vertical reflection: 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 vertical shift, THEN a vertical reflection 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) } . However, if we apply the vertical reflection before the vertical shift, we will get 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 h(x) = -x^3 + 3 } , which is not equal to our previous 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) } .

Resources

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