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]
