If [tex][tex]$y = x + 2$[/tex][/tex] were changed to [tex][tex]$y = x - 1$[/tex][/tex], how would the graph of the new function compare with the first one?

A. It would be shifted up.
B. It would be less steep.
C. It would be shifted down.
D. It would be shifted left.



Answer :

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.