Which translation maps the graph of the function [tex]f(x)=x^2[/tex] onto the function [tex]g(x)=x^2-6 x+6[/tex] ?
A. left 3 units, down 3 units
B. right 3 units, down 3 units
C. left 6 units, down 1 unit
D. right 6 units, down 1 unit
Answer (2)
Complete the square for g ( x ) = x 2 − 6 x + 6 to get g ( x ) = ( x − 3 ) 2 − 3 .
Identify the vertex of f ( x ) = x 2 as ( 0 , 0 ) and the vertex of g ( x ) = ( x − 3 ) 2 − 3 as ( 3 , − 3 ) .
Determine the horizontal shift as 3 units to the right and the vertical shift as 3 units down.
Conclude that the translation is right 3 units, down 3 units, so the answer is right 3 units, down 3 units .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 and we want to find the translation that maps it to the function g ( x ) = x 2 − 6 x + 6 . To find this translation, we need to rewrite g ( x ) in vertex form, which is g ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.
Completing the Square To rewrite g ( x ) = x 2 − 6 x + 6 in vertex form, we complete the square. We have
g ( x ) = x 2 − 6 x + 6
g ( x ) = ( x 2 − 6 x ) + 6
To complete the square, we need to add and subtract ( 2 − 6 ) 2 = ( − 3 ) 2 = 9 inside the parenthesis:
g ( x ) = ( x 2 − 6 x + 9 − 9 ) + 6
g ( x ) = ( x 2 − 6 x + 9 ) − 9 + 6
g ( x ) = ( x − 3 ) 2 − 3
So, the vertex form of g ( x ) is ( x − 3 ) 2 − 3 .
Finding the Translation The vertex of f ( x ) = x 2 is ( 0 , 0 ) . The vertex of g ( x ) = ( x − 3 ) 2 − 3 is ( 3 , − 3 ) .
To map the graph of f ( x ) to the graph of g ( x ) , we need to translate the vertex from ( 0 , 0 ) to ( 3 , − 3 ) . This means a horizontal shift of 3 units to the right and a vertical shift of 3 units down.
Conclusion Therefore, the translation that maps the graph of f ( x ) = x 2 onto the graph of g ( x ) = x 2 − 6 x + 6 is a translation of 3 units to the right and 3 units down.
Examples
Understanding translations of functions is useful in many fields. For example, in physics, understanding how to translate a function can help describe the motion of an object. If you know the initial position of an object as a function of time, you can use translations to determine its position at a later time or under different conditions. Similarly, in computer graphics, translations are used to move objects around the screen. By understanding how to translate functions, you can easily manipulate and animate objects in a virtual environment. For example, the function f ( x ) = x 2 can be translated to g ( x ) = ( x − 3 ) 2 − 3 , which shifts the parabola 3 units to the right and 3 units down.
The translation that maps the function f ( x ) = x 2 onto g ( x ) = x 2 − 6 x + 6 involves moving the graph 3 units to the right and 3 units down. This is determined by rewriting g ( x ) in vertex form, which reveals the new vertex to be at ( 3 , − 3 ) . Therefore, the correct option is B.
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