Select the correct answer.
Consider the function [tex]f(x)=10^x[/tex] and the function [tex]g(x)[/tex], which is shown below. How will the graph of [tex]g(x)[/tex] differ from the graph of [tex]f(x)[/tex]? [tex]g(x)=f(x-6)=10^{(x-6)}[/tex]
A. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] shifted to the left 6 units.
B. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] shifted to the right 6 units.
C. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] shifted 6 units up.
D. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] shifted 6 units down.
Answer (1)
The problem involves a horizontal shift of a function.
Replacing x with ( x − 6 ) in f ( x ) shifts the graph horizontally.
f ( x − 6 ) represents a shift to the right by 6 units.
The graph of g ( x ) is the graph of f ( x ) shifted to the right 6 units, so the answer is B. B
Explanation
Understanding the Problem We are given two functions, f ( x ) = 1 0 x and g ( x ) = f ( x − 6 ) = 1 0 ( x − 6 ) . We want to determine how the graph of g ( x ) differs from the graph of f ( x ) . This involves understanding transformations of functions, specifically horizontal shifts.
Identifying the Transformation The function g ( x ) = f ( x − 6 ) represents a horizontal shift of the function f ( x ) . When we replace x with ( x − 6 ) in the function f ( x ) , it shifts the graph horizontally.
Determining the Direction of the Shift Since we have f ( x − 6 ) , this means the graph of f ( x ) is shifted to the right by 6 units. To see why, consider a point on the graph of f ( x ) , say ( a , f ( a )) . The corresponding point on the graph of g ( x ) will be ( a + 6 , f ( a )) , because g ( a + 6 ) = f (( a + 6 ) − 6 ) = f ( a ) . Thus, the graph of g ( x ) is the graph of f ( x ) shifted to the right by 6 units.
Conclusion Therefore, the graph of g ( x ) is the graph of f ( x ) shifted to the right 6 units.
Examples
Imagine you are designing a website and want to animate elements moving across the screen. The function f ( x ) = 1 0 x could represent the initial position of an element, and g ( x ) = 1 0 ( x − 6 ) could represent the element's position after a delay or shift in time. Understanding how to shift functions horizontally allows you to control the timing and placement of these animations, creating visually appealing and interactive user experiences. This concept is fundamental in animation and interactive design, where precise control over movement and timing is crucial.
