Respuesta :
We will investigate the effect of translation on a given function f ( x ).
Translation is a type of transformation that deals with a given function f ( x ) in such a way that it displaces the entire function in four possible directions: up,down,left and right.
The number of units a function is to be translated in any direction is given by values of some characteristic constant.
The translated function can be expressed in a generalized form:
[tex]f^{\cdot}(x)\text{ = f ( x + a ) + b}[/tex]Where,
[tex]\begin{gathered} a\colon\text{ Magnitude of Horizontal Translation} \\ b\colon\text{ Magnitude of Vertical Translation} \end{gathered}[/tex]Each of the characteristic constant of translation ( a and b ) can be used to determine the direction of translation. The guidelines that are used to express ( a and b ) are:
[tex]\begin{gathered} a\text{ > 0 }\ldots\text{ Left translation} \\ a\text{ < 0 }\ldots\text{ Right translation} \\ \\ b\text{ > 0 }\ldots\text{ Upwards} \\ b\text{ < 0 }\ldots\text{ Downwards} \end{gathered}[/tex]The signs of each constant determine the exact direction of translation.
We are given a function f ( x ) as follows:
[tex]y\text{ = }\sqrt[]{x}[/tex]We are asked to find the new function such that the original function ( y ) has been translated ( 4 ) units to the left.
Using the above guidelines we can say that we are undergoing only horizontal translation; hence:
[tex]a\text{ }\ne\text{ 0 and b = 0}[/tex]So the general form is reduced down to:
[tex]y\text{ = f ( x + a )}[/tex]To determine the value and sign of characteristic constant ( a ) we will use the next set of guidelines. All left translations are accompanied by a positive value of ( a ). Hence,
[tex]a\text{ > 0 }\ldots\text{ Left translation}[/tex]The magnitude of the left translation given is ( 4 units ). Hence, the value of the characteristic constant is:
[tex]a\text{ = 4}[/tex]Then the generalized function depiction would be:
[tex]y\text{ = f ( x + 4 )}[/tex]We will substitute whatever is within the parenthesis of the f ( x + 4 ) into the given function ( x ) as follows:
[tex]x\to\text{ (x+4)}[/tex]Then the resulting translated function would be expressed as:
[tex]\textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{\sqrt[]{(x+4)}\ldots}\text{\textcolor{#FF7968}{ Answer ( Option B )}}[/tex]