Let's compare the graph of [tex]\( g(x) = x - 3 \)[/tex] with the graph of the parent function [tex]\( f(x) = x \)[/tex].
1. Understanding the Parent Function [tex]\( f(x) = x \)[/tex]:
- The graph of [tex]\( f(x) = x \)[/tex] is a straight line passing through the origin [tex]\((0, 0)\)[/tex] with a slope of 1.
- It is the simplest linear function, where for any value of [tex]\( x \)[/tex], the output [tex]\( y \)[/tex] is identical to [tex]\( x \)[/tex].
2. Understanding the Function [tex]\( g(x) = x - 3 \)[/tex]:
- The function [tex]\( g(x) = x - 3 \)[/tex] takes the form of a linear function similar to [tex]\( f(x) = x \)[/tex], but it has a constant term [tex]\(-3\)[/tex].
- This constant term [tex]\(-3\)[/tex] indicates a vertical shift of the graph. Specifically, subtracting 3 from the output [tex]\( y \)[/tex]-value.
3. Effect on the Graph:
- For every [tex]\( x \)[/tex]-value, the output [tex]\( y \)[/tex]-value for [tex]\( g(x) \)[/tex] is 3 units less than the corresponding output for [tex]\( f(x) \)[/tex].
- Mathematically, if the coordinate point [tex]\((x, y)\)[/tex] lies on the graph of [tex]\( f(x) \)[/tex], then the point [tex]\((x, y-3)\)[/tex] will lie on the graph of [tex]\( g(x) \)[/tex].
4. Conclusion Based on the Comparison:
- This vertical shift does not affect the slope or the horizontal compression of the line; it only shifts each point of the parent function down by 3 units without altering the overall shape of the graph.
- Thus, we can confirm that:
- There is no horizontal compression.
- There is no vertical compression.
- There is no horizontal shift.
Therefore, the correct comparison is:
The graph of [tex]\( g(x) \)[/tex] is shifted 3 units down compared to the graph of the parent function [tex]\( f(x) = x \)[/tex].
The correct answer is:
B. The graph of [tex]\( g(x) \)[/tex] is shifted 3 units down.
