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

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



Answer :

To compare how the graph of the new function [tex]\( y = x - 1 \)[/tex] compares with the original function [tex]\( y = x + 2 \)[/tex], we need to look at how their equations differ and what that implies about their graphs.

1. Identify and Compare the Functions:
- The original function is [tex]\( y = x + 2 \)[/tex].
- The new function is [tex]\( y = x - 1 \)[/tex].

2. Analyze the Changes:
- Both functions are linear and of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In [tex]\( y = x + 2 \)[/tex], the slope [tex]\( m = 1 \)[/tex] and the y-intercept [tex]\( b = 2 \)[/tex].
- In [tex]\( y = x - 1 \)[/tex], the slope [tex]\( m = 1 \)[/tex] and the y-intercept [tex]\( b = -1 \)[/tex].

3. Compare Slopes:
- Both functions have the same slope, [tex]\( m = 1 \)[/tex]. This means the lines are parallel and equally steep.

4. Compare Intercepts:
- The original function intersects the y-axis at [tex]\( y = 2 \)[/tex].
- The new function intersects the y-axis at [tex]\( y = -1 \)[/tex].

5. Determine the Transformation:
- Since both lines have the same slope, the only difference between the two graphs is the vertical shift due to the difference in y-intercepts.
- The y-intercept changes from 2 to -1. This represents a shift down by 3 units (because [tex]\( -1 - 2 = -3 \)[/tex]).

Therefore, the graph of the new function [tex]\( y = x - 1 \)[/tex] is obtained by shifting the graph of [tex]\( y = x + 2 \)[/tex] down by 3 units.

The correct answer is:
A. It would be shifted down.