To determine how the graph of the new function [tex]\( y = x - 1 \)[/tex] compares with the original function [tex]\( y = x + 2 \)[/tex], let's consider the characteristics of each function.
The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
For the original function [tex]\( y = x + 2 \)[/tex]:
- The slope [tex]\( m \)[/tex] is 1.
- The y-intercept [tex]\( b \)[/tex] is 2, which means the graph intersects the y-axis at the point (0, 2).
For the new function [tex]\( y = x - 1 \)[/tex]:
- The slope [tex]\( m \)[/tex] is still 1.
- The y-intercept [tex]\( b \)[/tex] is -1, which means the graph intersects the y-axis at the point (0, -1).
Since both functions have the same slope, their steepness does not change. The difference lies in their y-intercepts:
- The original function [tex]\( y = x + 2 \)[/tex] intersects the y-axis at (0, 2).
- The new function [tex]\( y = x - 1 \)[/tex] intersects the y-axis at (0, -1).
This means that the entire graph of [tex]\( y = x - 1 \)[/tex] is shifted vertically downward by 3 units compared to the graph of [tex]\( y = x + 2 \)[/tex]. It is not shifted horizontally, nor has its steepness changed.
Thus, the correct answer is:
C. It would be shifted down.
