Home › Functions and graphs

Reflect and shift a square-root graph step by step

Outside changes y; inside changes x

A minus sign outside $f (x)$ reflects a graph across the $x$ -axis. Adding a constant outside shifts it vertically. Replacing $x$ by $x - h$ shifts it right by $h$ . Apply changes in the order stated and check the new starting point.

Problem

Starting from $y = x$ , reflect the graph across the $x$ -axis, shift it up $8$ units, then shift it right $9$ units. Give the final equation and its domain.

Answer

y = - x - 9 + 8

, with domain

[9, \infty)

The original starting point (0, 0) becomes (9, 8).

Step-by-step solution

1. Reflect

Reflecting $y = x$ across the $x$ -axis gives $y = - x$ .

2. Shift upward

Add $8$ outside the root: $y = - x + 8$ .

3. Shift right

Replace $x$ by $x - 9$ to move the graph right $9$ units:

y = - x - 9 + 8.

The expression under the square root must be nonnegative, so $x - 9 \geq 0$ and the domain is $[9, \infty)$ . At $x = 9$ , $y = 8$ , confirming the moved starting point.

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.