If you are given the graph of [tex]g(x)=\log _2 x[/tex], how could you graph [tex]f(x)=\log _2 x+5[/tex]?
A. Translate each point of the graph of [tex]g(x)[/tex] 5 units up.
B. Translate each point of the graph of [tex]g(x)[/tex] 5 units down.
C. Translate each point of the graph of [tex]g(x)[/tex] 5 units left.
D. Translate each point of the graph of [tex]g(x)[/tex] 5 units right.
Answer (2)
Adding a constant to a function results in a vertical translation.
A positive constant translates the graph upwards.
The graph of f ( x ) = lo g 2 x + 5 is the graph of g ( x ) = lo g 2 x translated 5 units up.
Therefore, translate each point of the graph of g ( x ) 5 units up.
Explanation
Understanding the Problem We are given the function $g(x) =
log_2 x an d w e w an tt o g r a p h t h e f u n c t i o n f(x) =
log_2 x + 5 . W e n ee d t o d e t er min e h o wt h e g r a p h o f f(x) i sre l a t e d t o t h e g r a p h o f g(x)$.
Vertical Translation The function f ( x ) is obtained by adding a constant, 5, to the function g ( x ) . Adding a constant to a function results in a vertical translation of the graph of the function.
Direction and Magnitude Since we are adding a positive constant, the translation is upwards. The magnitude of the translation is equal to the constant, which is 5. Therefore, the graph of f ( x ) is obtained by translating the graph of g ( x ) 5 units upwards.
Graphing f(x) To graph $f(x) =
log_2 x + 5 , w e t ak ee a c h p o in t o n t h e g r a p h o f g(x) =
log_2 x$ and move it 5 units upwards.
Examples
Imagine you are designing a website and you want to position an element 5 pixels lower than its original position. This is a vertical translation. Similarly, in physics, if you are analyzing the potential energy of an object and you add a constant to the potential energy function, you are effectively shifting the entire potential energy curve vertically. Understanding vertical translations is useful in various fields, including computer graphics, physics, and engineering.
To graph f ( x ) = lo g 2 x + 5 , we need to translate the graph of g ( x ) = lo g 2 x upwards by 5 units. This means that every point on the graph of g ( x ) will be shifted 5 units higher, resulting in the graph of f ( x ) . Therefore, the correct option is A: Translate each point of the graph of g ( x ) 5 units up.
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