The function [tex]f(x)=x^2[/tex] has been translated 9 units up and 4 units to the right to form the function [tex]g(x)[/tex]. Which represents [tex]g(x)[/tex]?
A. [tex]g(x)=(x+9)^2+4[/tex]
B. [tex]g(x)=(x+9)^2-4[/tex]
C. [tex]g(x)=(x-4)^2+9[/tex]
D. [tex]g(x)=(x+4)^2+9[/tex]
Answer (1)
To shift the function f ( x ) = x 2 to the right by 4 units, replace x with ( x − 4 ) , resulting in ( x − 4 ) 2 .
To shift the function 9 units up, add 9 to the function, resulting in ( x − 4 ) 2 + 9 .
Therefore, the translated function is g ( x ) = ( x − 4 ) 2 + 9 .
The final answer is ( x − 4 ) 2 + 9 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 and asked to find the function g ( x ) which is the result of translating f ( x ) 9 units up and 4 units to the right.
Horizontal Translation To translate a function f ( x ) to the right by h units, we replace x with ( x − h ) . In this case, we are translating f ( x ) 4 units to the right, so we replace x with ( x − 4 ) to get ( x − 4 ) 2 .
Vertical Translation To translate a function f ( x ) up by k units, we add k to the function. In this case, we are translating the function 9 units up, so we add 9 to the result from the horizontal translation. This gives us g ( x ) = ( x − 4 ) 2 + 9 .
Final Answer Therefore, the function g ( x ) is represented by ( x − 4 ) 2 + 9 .
Examples
Imagine you're designing a skateboarding ramp. The basic shape is a parabola described by f ( x ) = x 2 . To make the ramp more interesting, you shift it 4 feet to the right and raise it 9 feet higher. The new shape of the ramp is now described by g ( x ) = ( x − 4 ) 2 + 9 . This transformation allows you to adjust the ramp's position and height to fit your design needs, making it both functional and fun.
