To determine how the graph of the function [tex]\( g(x) \)[/tex] differs from the graph of the function [tex]\( f(x) \)[/tex], we need to closely examine the relationship between [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex].
Given:
[tex]\[ f(x) = 2^x \][/tex]
[tex]\[ g(x) = f(x) + 6 \][/tex]
Substituting [tex]\( f(x) \)[/tex] into the equation for [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x + 6 \][/tex]
Now let’s analyze what this means graphically:
1. Vertical Shift: When a constant is added to a function, it results in a vertical shift. Here, adding 6 to [tex]\( 2^x \)[/tex] means taking every point on the graph of [tex]\( f(x) \)[/tex] and shifting it vertically up by 6 units. This is because for every value of [tex]\( x \)[/tex], the value of [tex]\( g(x) \)[/tex] is exactly 6 units more than the value of [tex]\( f(x) \)[/tex] at that same [tex]\( x \)[/tex].
2. Horizontal Shift: A horizontal shift, either left or right, occurs when the input [tex]\( x \)[/tex] of the function is altered. For example, [tex]\( f(x-h) \)[/tex] shifts the graph to the right by [tex]\( h \)[/tex] units, and [tex]\( f(x+h) \)[/tex] shifts the graph to the left by [tex]\( h \)[/tex] units. Here, we do not have such an alteration in the input [tex]\( x \)[/tex]; therefore, there is no horizontal shift in this case.
3. Direction of the Shift: Adding a positive constant (like 6) results in a shift upward. Adding a negative constant would result in a shift downward. Basic principles of transformations confirm this.
Given these points, the correct description of the transformation is that the graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 6 units up.
Thus, the correct answer is:
B. The graph of function [tex]\( g \)[/tex] is the graph of function [tex]\( f \)[/tex] shifted 6 units up.
