Answer :
To determine the key features of the function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex], we will analyze the function step by step:
1. Vertex at (8, -1):
The function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] is in the form of [tex]\( f(x) = a|b(x-h)| + k \)[/tex], where the vertex is given by the point [tex]\((h, k)\)[/tex]. Here, [tex]\( h = 8 \)[/tex] and [tex]\( k = -1 \)[/tex]. Therefore, the vertex of the function is at [tex]\((8, -1)\)[/tex].
- This statement is correct.
2. No [tex]\(x\)[/tex]-intercepts:
To find the [tex]\(x\)[/tex]-intercepts, we set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ |4(x-8)| - 1 = 0 \implies |4(x-8)| = 1 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\(4(x-8) = 1\)[/tex] or [tex]\(4(x-8) = -1\)[/tex]:
[tex]\[ x - 8 = \frac{1}{4} \implies x = 8.25 \][/tex]
[tex]\[ x - 8 = -\frac{1}{4} \implies x = 7.75 \][/tex]
Since the function has two [tex]\(x\)[/tex]-intercepts, the statement that there are no [tex]\(x\)[/tex]-intercepts is incorrect.
- This statement is incorrect.
3. Symmetric about the line [tex]\( x = 8 \)[/tex]:
The function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] is an absolute value function centered at [tex]\( x = 8 \)[/tex]. Therefore, it is symmetric about the line [tex]\( x = 8 \)[/tex].
- This statement is correct.
4. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity:
An absolute value function such as [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] will always have non-negative values inside the absolute value. The value inside the absolute value gets large and positive or negative but never approaches negative infinity; instead, [tex]\( f(x) \)[/tex] approaches positive infinity. Therefore, this statement is incorrect.
- This statement is incorrect.
5. Domain of [tex]\( [8, \infty) \)[/tex]:
The domain of the function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] is all real numbers since the absolute value function is defined for all [tex]\( x \)[/tex]. Hence, the domain is not limited to starting from 8.
- This statement is incorrect.
6. Range of [tex]\([-1, \infty)\)[/tex]:
Since the minimum value of [tex]\( |4(x-8)| \)[/tex] is 0, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( -1 \)[/tex], and it can go up to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] moves away from 8. Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\([-1, \infty)\)[/tex].
- This statement is correct.
Based on the analysis, the correct key features of the function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] are:
- Vertex at [tex]\((8, -1)\)[/tex]
- Symmetric about the line [tex]\( x = 8 \)[/tex]
- Range of [tex]\([-1, \infty)\)[/tex]
The numerical result is thus: [tex]\[ [1, 0, 1, 0, 0, 1] \][/tex].
1. Vertex at (8, -1):
The function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] is in the form of [tex]\( f(x) = a|b(x-h)| + k \)[/tex], where the vertex is given by the point [tex]\((h, k)\)[/tex]. Here, [tex]\( h = 8 \)[/tex] and [tex]\( k = -1 \)[/tex]. Therefore, the vertex of the function is at [tex]\((8, -1)\)[/tex].
- This statement is correct.
2. No [tex]\(x\)[/tex]-intercepts:
To find the [tex]\(x\)[/tex]-intercepts, we set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ |4(x-8)| - 1 = 0 \implies |4(x-8)| = 1 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\(4(x-8) = 1\)[/tex] or [tex]\(4(x-8) = -1\)[/tex]:
[tex]\[ x - 8 = \frac{1}{4} \implies x = 8.25 \][/tex]
[tex]\[ x - 8 = -\frac{1}{4} \implies x = 7.75 \][/tex]
Since the function has two [tex]\(x\)[/tex]-intercepts, the statement that there are no [tex]\(x\)[/tex]-intercepts is incorrect.
- This statement is incorrect.
3. Symmetric about the line [tex]\( x = 8 \)[/tex]:
The function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] is an absolute value function centered at [tex]\( x = 8 \)[/tex]. Therefore, it is symmetric about the line [tex]\( x = 8 \)[/tex].
- This statement is correct.
4. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity:
An absolute value function such as [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] will always have non-negative values inside the absolute value. The value inside the absolute value gets large and positive or negative but never approaches negative infinity; instead, [tex]\( f(x) \)[/tex] approaches positive infinity. Therefore, this statement is incorrect.
- This statement is incorrect.
5. Domain of [tex]\( [8, \infty) \)[/tex]:
The domain of the function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] is all real numbers since the absolute value function is defined for all [tex]\( x \)[/tex]. Hence, the domain is not limited to starting from 8.
- This statement is incorrect.
6. Range of [tex]\([-1, \infty)\)[/tex]:
Since the minimum value of [tex]\( |4(x-8)| \)[/tex] is 0, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( -1 \)[/tex], and it can go up to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] moves away from 8. Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\([-1, \infty)\)[/tex].
- This statement is correct.
Based on the analysis, the correct key features of the function [tex]\( f(x) = |4(x-8)| - 1 \)[/tex] are:
- Vertex at [tex]\((8, -1)\)[/tex]
- Symmetric about the line [tex]\( x = 8 \)[/tex]
- Range of [tex]\([-1, \infty)\)[/tex]
The numerical result is thus: [tex]\[ [1, 0, 1, 0, 0, 1] \][/tex].
