To understand how the function [tex]\( h(x) = g(x) + 1 \)[/tex] transforms the graph of [tex]\( g(x) = x^2 \)[/tex], we first need to analyze the effects of the transformation.
Consider the function [tex]\( g(x) = x^2 \)[/tex]. This is a simple quadratic function whose graph is a parabola opening upwards with its vertex at the origin (0,0).
Now, let's consider the transformation to obtain [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = g(x) + 1 \][/tex]
This can be explicitly written in terms of [tex]\( x \)[/tex] as:
[tex]\[ h(x) = x^2 + 1 \][/tex]
This transformation adds 1 to the output (the [tex]\( y \)[/tex]-value) of the original function [tex]\( g(x) \)[/tex]. Such a transformation affects the vertical positioning of the graph. Specifically:
- Adding a positive constant to the function [tex]\( g(x) \)[/tex] results in the entire graph of [tex]\( g(x) \)[/tex] being shifted vertically upward by that constant.
In this case, since the constant added is 1, the graph of [tex]\( g(x) \)[/tex] will be shifted up by 1 unit. Hence, for every [tex]\( x \)[/tex], the value of [tex]\( h(x) \)[/tex] will be 1 unit higher than that of [tex]\( g(x) \)[/tex].
Therefore, the correct statement is:
C. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted up 1 unit.
