Let's examine how the graph of the function [tex]\( g(x) = x^2 \)[/tex] is transformed to obtain the function [tex]\( h(x) = g(x) - 5 \)[/tex].
1. Original Function:
The function [tex]\( g(x) = x^2 \)[/tex] is a basic quadratic function, which is a parabola opening upwards with its vertex at the origin (0, 0).
2. Transformation Applied:
The transformation [tex]\( h(x) = g(x) - 5 \)[/tex] can be interpreted as [tex]\( h(x) = x^2 - 5 \)[/tex].
3. Effect of the Transformation:
Subtracting 5 from the function [tex]\( g(x) \)[/tex] results in a vertical shift of the graph:
- When we say [tex]\( g(x) = x^2 - 5 \)[/tex], this means we are moving every point on the graph of [tex]\( g(x) \)[/tex] down by 5 units.
- For example, the vertex of the parabola [tex]\( g(x) = x^2 \)[/tex], which is at (0, 0), will move to (0, -5).
4. Conclusion:
The function [tex]\( h(x) = g(x) - 5 \)[/tex] is a vertical shift of the function [tex]\( g(x) = x^2 \)[/tex] downwards by 5 units.
Hence, the correct answer is:
D. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] vertically shifted down 5 units.
