Question

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

Answer 1

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

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

To construct a graph of `f(x)=3|x+5|`, we have to take the right side (for `x>=-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]}