Chapter 1.6, Problem 59E

Solution

The statement “The function $y=f\left(x+3\right)$ represent a translation to the right by $3$ units of the graph of $y=f\left(x\right)$ ” is true or false..

The statement is false.

Given:

The statement “The function $y=f\left(x+3\right)$ represent a translation to the right by $3$ units of the graph of $y=f\left(x\right)$ ”.

Concept Used:

The graph of $y=f\left(x\right)$ is translated or shifted horizontally left or right according as:

1) If $y=f\left(x-c\right)$ , then graph shifted by c units right from $y=f\left(x\right)$ .

2) If $y=f\left(x+c\right)$ , then graph shifted by c units left from $y=f\left(x\right)$ .

And, the graph of $y=f\left(x\right)$ is translated or shifted vertically up or down according as:

1) If $y=f\left(x\right)+c$ , then graph shifted by c units up from $y=f\left(x\right)$ .

2) If $y=f\left(x\right)-c$ , then graph shifted by c units down from $y=f\left(x\right)$ .

Calculation:

Consider the given statement says, “The function $y=f\left(x+3\right)$ represent a translation to the right by $3$ units of the graph of $y=f\left(x\right)$ ”

Since, the translation $y=f\left(x+c\right)$ translates the graph of $y=f\left(x\right)$ by $c$ units left, which contradict the given statement. Hence, the given statement is false.

Conclusion The statement is false.