# How do you graph f(x)=3|x+5|?

Jun 12, 2016

See the explanation below

#### Explanation:

Recall the definition of the absolute value of a real number.

Absolute value of a non-negative number is this number itself, which can be expressed as
IF $R \ge 0$ THEN $| R | = R$
Absolute value of a negative number is the negation of this number, which can be expressed as
IF $R < 0$ THEN $| R | = - R$

Applying this to our function $f \left(x\right) = 3 | x + 5 |$, we can say that for all $x$ that satisfy $x + 5 \ge 0$ (that is, $x \ge - 5$) we have $| x + 5 | = x + 5$, and our function is equivalent to $f \left(x\right) = 3 \left(x + 5\right)$.
Its graph looks like
graph{3(x+5) [-10, 10, -5, 5]}

For all other $x$ (that is, $x < - 5$) we have $| x + 5 | = - \left(x + 5\right)$, and our function is equivalent to $f \left(x\right) = - 3 \left(x + 5\right)$.
Is graph looks like
graph{-3(x+5) [-10, 10, -5, 5]}

To construct a graph of $f \left(x\right) = 3 | x + 5 |$, we have to take the right side (for $x \ge - 5$) from the first graph and the left side (for $x < - 5$) from the second graph. The result is

graph{3|x+5| [-10, 10, -5, 5]}