You are given the graph of [tex]f(x)=\log _6 x[/tex], how could you graph
A. Translate each point of the graph of [tex]h(x)[/tex] 3 units up.
B. Translate each point of the graph of [tex]h(x)[/tex] 3 units down.
C. Translate each point of the graph of [tex]h(x)[/tex] 3 units right.
D. Translate each point of the graph of [tex]h(x)[/tex] 3 units left.
Answer (1)
To translate the graph of h ( x ) 3 units up, we have y = h ( x ) + 3 .
To translate the graph of h ( x ) 3 units down, we have y = h ( x ) − 3 .
To translate the graph of h ( x ) 3 units to the right, we have y = h ( x − 3 ) .
To translate the graph of h ( x ) 3 units to the left, we have y = h ( x + 3 ) .
Explanation
Understanding the Problem We are given the graph of a function f ( x ) = lo g 6 x and we want to describe how to graph transformations of a general function h ( x ) . Specifically, we want to describe translations of the graph of h ( x ) by 3 units up, 3 units down, 3 units right, and 3 units left.
Translating Up To translate the graph of h ( x ) 3 units up, we add 3 to the function value. This means that the new function is y = h ( x ) + 3 . Every point ( x , y ) on the graph of h ( x ) is transformed to ( x , y + 3 ) on the translated graph.
Translating Down To translate the graph of h ( x ) 3 units down, we subtract 3 from the function value. This means that the new function is y = h ( x ) − 3 . Every point ( x , y ) on the graph of h ( x ) is transformed to ( x , y − 3 ) on the translated graph.
Translating Right To translate the graph of h ( x ) 3 units to the right, we replace x with ( x − 3 ) in the function. This means that the new function is y = h ( x − 3 ) . Every point ( x , y ) on the graph of h ( x ) is transformed to ( x + 3 , y ) on the translated graph.
Translating Left To translate the graph of h ( x ) 3 units to the left, we replace x with ( x + 3 ) in the function. This means that the new function is y = h ( x + 3 ) . Every point ( x , y ) on the graph of h ( x ) is transformed to ( x − 3 , y ) on the translated graph.
Final Answer In summary:
To translate the graph of h ( x ) 3 units up, the new function is y = h ( x ) + 3 .
To translate the graph of h ( x ) 3 units down, the new function is y = h ( x ) − 3 .
To translate the graph of h ( x ) 3 units to the right, the new function is y = h ( x − 3 ) .
To translate the graph of h ( x ) 3 units to the left, the new function is y = h ( x + 3 ) .
Examples
Imagine you are designing a video game and you want to move a character's position on the screen. If the character's initial position is described by a function h ( x ) , then shifting the character up by 3 units would be represented by h ( x ) + 3 , shifting it down by 3 units would be h ( x ) − 3 , shifting it to the right by 3 units would be h ( x − 3 ) , and shifting it to the left by 3 units would be h ( x + 3 ) . Understanding these transformations allows you to precisely control the character's movement in the game.
