Consider the function [tex]f(x)=2^x[/tex] and the function [tex]g(x)[/tex].
[tex]g(x)=f(x)+4=2^x+4[/tex]
How will the graph of [tex]g(x)[/tex] differ from the graph of [tex]f(x)[/tex]?
A. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] vertically compressed 4 units. B. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] shifted 4 units downward. C. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] vertically stretched 4 units. D. The graph of [tex]g(x)[/tex] is the graph of [tex]f(x)[/tex] shifted 4 units upward.
To determine how the graph of [tex]\( g(x) \)[/tex] differs from the graph of [tex]\( f(x) \)[/tex], let's examine the given functions: [tex]\[ f(x) = 2^x \][/tex] [tex]\[ g(x) = f(x) + 4 = 2^x + 4 \][/tex]
The function [tex]\( g(x) \)[/tex] takes the original function [tex]\( f(x) \)[/tex] and adds 4 to it. This addition is applied to the output of [tex]\( f(x) \)[/tex], meaning that for any input [tex]\( x \)[/tex], the output of [tex]\( g(x) \)[/tex] is 4 units greater than the output of [tex]\( f(x) \)[/tex].
This can be visualized as a vertical shift: - Adding a constant to a function results in a vertical shift of the graph. - In this case, [tex]\( f(x) \)[/tex] is shifted vertically by [tex]\( +4 \)[/tex].
Therefore, the graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 4 units upward.
The correct answer is: D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] shifted 4 units upward.
