Question

vertical shifts quick check
item 1
which statement best shows how the vertical shift affects the parent function ( y = x ) when shifted to ( y = x + 5 )? (1 point)
( y = x + 5 ) will have each point 5 steps higher than ( y = x ).
item 2
( y = x + 5 ) will have each point 5 steps to the right of ( y = x ).
item 3
( y = x + 5 ) will have each point 5 steps lower than ( y = x ).
item 4
( y = x + 5 ) will have each point 5 steps to the left of ( y = x ).
item 5

Explanation:

Step1: Recall Vertical Shift Rule

For a function \( y = f(x)+k \), if \( k>0 \), the graph of \( y = f(x) \) is shifted up \( k \) units. Here, the parent function is \( y = x \) (i.e., \( f(x)=x \)) and the transformed function is \( y=x + 5 \) (so \( k = 5>0 \)).

Step2: Analyze Each Option

• Option 1: " \( y=x + 5 \) will have each point 5 steps higher than \( y = x \)." This matches the vertical shift rule (upward shift by 5 units means each point's \( y \)-coordinate is 5 more, so the point is 5 steps higher).
• Option 2: " \( y=x + 5 \) will have each point 5 steps lower than \( y = x \)." This is incorrect as adding 5 to \( y \)-coordinate (from \( y=x \) to \( y=x + 5 \)) increases the height, not decreases.
• Option 3: " \( y=x + 5 \) will have each point 5 steps to the right of \( y = x \)." Horizontal shifts are of the form \( y=f(x - h) \), not vertical shifts. So this is about horizontal shift, not vertical, so incorrect.
• Option 4: " \( y=x + 5 \) will have each point 5 steps to the left of \( y = x \)." Again, this is a horizontal shift concept (left shift is \( y=f(x + h) \)), not vertical, so incorrect.

Answer:

\( y = x + 5 \) will have each point 5 steps higher than \( y = x \) (the first option among the given choices for \( y = x+5 \) effect on \( y = x \))