What is the vertex of [tex]$f(x)=|x+8|-3$[/tex]?

A. [tex](-8, -3)[/tex]
B. [tex](-8, 3)[/tex]
C. [tex](8, -3)[/tex]
D. [tex](8, 3)[/tex]



Answer :

To find the vertex of the function [tex]\( f(x)=|x+8|-3 \)[/tex], let's break it down step by step:

1. Identify the inside expression of the absolute value function:

The given function is [tex]\( f(x) = |x + 8| - 3 \)[/tex]. Here, the expression inside the absolute value is [tex]\( x + 8 \)[/tex].

2. Determine where the expression inside the absolute value is zero:

To find the vertex, we need to determine the point where [tex]\( x + 8 = 0 \)[/tex].

Solving for [tex]\( x \)[/tex]:
[tex]\[ x + 8 = 0 \implies x = -8 \][/tex]

So, the [tex]\( x \)[/tex]-coordinate of the vertex is [tex]\( -8 \)[/tex].

3. Evaluate the function at this [tex]\( x \)[/tex]-coordinate to find the [tex]\( y \)[/tex]-coordinate:

Substitute [tex]\( x = -8 \)[/tex] into the function:
[tex]\[ f(-8) = | -8 + 8 | - 3 = | 0 | - 3 = 0 - 3 = -3 \][/tex]

So, the [tex]\( y \)[/tex]-coordinate of the vertex is [tex]\( -3 \)[/tex].

4. Combine the results to find the vertex:

The vertex is the point where the function reaches its minimum value, and given our results, this point is:
[tex]\[ (-8, -3) \][/tex]

So, the vertex of the function [tex]\( f(x)=|x+8|-3 \)[/tex] is indeed [tex]\( (-8, -3) \)[/tex].

Among the given options, the correct choice is:
[tex]\[ (-8, -3) \][/tex]