Let's analyze the function [tex]\( f(x) = |4(x - 8)| - 1 \)[/tex] to determine its key features.
1. Vertex:
- The vertex of the function [tex]\(|4(x - 8)| -1\)[/tex] occurs where the expression inside the absolute value is zero.
- So, set [tex]\(4(x - 8) = 0\)[/tex].
- Solving for [tex]\(x\)[/tex], we find [tex]\(x = 8\)[/tex].
- Substituting [tex]\(x = 8\)[/tex] into the function, we get [tex]\(f(8) = |4(8 - 8)| - 1 = 0 - 1 = -1\)[/tex].
- Therefore, the vertex of the function is at [tex]\((8, -1)\)[/tex].
2. Symmetry:
- The function [tex]\( f(x) = |4(x - 8)| - 1 \)[/tex] is symmetric about the line [tex]\(x = 8\)[/tex], as the absolute value function ensures symmetry around the point where [tex]\(4(x-8) = 0\)[/tex].
3. Range:
- The minimum value of the function is at the vertex where [tex]\(x = 8\)[/tex] and [tex]\(f(8) = -1\)[/tex].
- As [tex]\(x\)[/tex] moves away from [tex]\(8\)[/tex], [tex]\( |4(x - 8)| \)[/tex] increases, making [tex]\(f(x) = |4(x-8)| - 1\)[/tex] increase as well.
- Therefore, the range of the function is [tex]\([-1, \infty)\)[/tex].
4. Domain:
- The function [tex]\( f(x) = |4(x - 8)| - 1 \)[/tex] is defined for all real values of [tex]\(x\)[/tex], as the absolute value can handle any real input.
- Hence, the domain is all real numbers: [tex]\((-\infty, \infty)\)[/tex].
5. x-Intercepts:
- To find the x-intercepts, set [tex]\( f(x) = 0 \)[/tex].
- Solving [tex]\(|4(x - 8)| - 1 = 0\)[/tex], we get [tex]\(|4(x - 8)| = 1\)[/tex].
- This gives us two equations: [tex]\(4(x - 8) = 1\)[/tex] and [tex]\(4(x - 8) = -1\)[/tex].
- Solving these, we get [tex]\(x = 8.25\)[/tex] and [tex]\(x = 7.75\)[/tex].
- Therefore, the function does have x-intercepts at [tex]\(x = 8.25\)[/tex] and [tex]\(x = 7.75\)[/tex].
6. Behavior as [tex]\( x \)[/tex] approaches negative infinity:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( |4(x - 8)| \)[/tex] increases indefinitely, leading [tex]\( f(x) \)[/tex] to approach infinity, not negative infinity.
Given these analyses, we can conclude the correct attributes of the function [tex]\( f(x) = |4(x - 8)| - 1 \)[/tex]:
- Range of [tex]\([-1, \infty)\)[/tex].
- Vertex at [tex]\((8, -1)\)[/tex].
- Symmetric about the line [tex]\( x = 8 \)[/tex].
So the correct answers are:
- range of [tex]\([-1, \infty)\)[/tex]
- vertex at [tex]\((8, -1)\)[/tex]
- symmetric about the line [tex]\(x=8\)[/tex]
