Chapter 2.5, Problem 5AYU

To determine

Whether the graph of $y=\sqrt{x+2}$ is obtained by a horizontal shift of graph of $y=f\left(x\right)$ to the right at a distance 2 units.

Answer to Problem 5AYU

The statement is false.

Explanation of Solution

For $c>0$ .

  $h\left(x\right)=f\left(x\right)+c$ shifts the graph of $f\left(x\right)$ vertically upward by $c$ units,

  $h\left(x\right)=f\left(x\right)–c$ shifts the graph of $f\left(x\right)$ vertically downward by $c$ units,

  $h\left(x\right)=f\left(x+c\right)$ shifts the graph of $f\left(x\right)$ horizontally left by $c$ units,

  $h\left(x\right)=f\left(x–c\right)$ shifts the graph of $f\left(x\right)$ horizontally right by $c$ units.

Now, consider the graph of $y=\sqrt{x+2}$ .

Here, the graph is of the form $h\left(x\right)=f\left(x+c\right)$ ,

So, $c=2$

The graph of $y=\sqrt{x+2}$ can be obtained by a horizontal shift of the graph of $y=f\left(x\right)$ to the left at a distance 2 units.

So, the statement “the graph of $y=\sqrt{x+2}$ is obtained by a horizontal shift of graph of $y=f\left(x\right)$ to the right at a distance 2 units” is false.