Answer :
To determine which functions have a vertex with an [tex]$x$[/tex]-value of 0, we need to analyze the behavior of each absolute value function.
1. [tex]\( f(x) = |x| \)[/tex]
- The function [tex]\( f(x) = |x| \)[/tex] has its vertex at the point where the expression inside the absolute value is zero, which happens when [tex]\( x = 0 \)[/tex].
- Thus, the vertex is at [tex]\( (0,0) \)[/tex], and the [tex]\( x \)[/tex]-value of the vertex is indeed 0.
2. [tex]\( f(x) = |x| + 3 \)[/tex]
- The function [tex]\( f(x) = |x| + 3 \)[/tex] is a vertical shift of the function [tex]\( f(x) = |x| \)[/tex].
- The vertex of [tex]\( f(x) = |x| \)[/tex] is at [tex]\( (0,0) \)[/tex]. Adding 3 shifts this vertex up by 3 units but does not change the [tex]\( x \)[/tex]-value.
- Therefore, the vertex remains at [tex]\( x = 0 \)[/tex].
3. [tex]\( f(x) = |x+3| \)[/tex]
- The function [tex]\( f(x) = |x+3| \)[/tex] reaches its minimum when the expression inside the absolute value is zero, which occurs at [tex]\( x = -3 \)[/tex].
- Therefore, the vertex has an [tex]\( x \)[/tex]-value of [tex]\( -3 \)[/tex], not 0.
4. [tex]\( f(x) = |x| - 6 \)[/tex]
- The function [tex]\( f(x) = |x| - 6 \)[/tex] is a vertical shift of the function [tex]\( f(x) = |x| \)[/tex].
- The vertex of [tex]\( f(x) = |x| \)[/tex] is at [tex]\( (0,0) \)[/tex]. Subtracting 6 shifts this vertex down by 6 units but does not change the [tex]\( x \)[/tex]-value.
- Therefore, the vertex remains at [tex]\( x = 0 \)[/tex].
5. [tex]\( f(x) = |x+3| - 6 \)[/tex]
- Similar to [tex]\( f(x) = |x+3| \)[/tex], the [tex]\( x \)[/tex]-value of the vertex will be determined by the expression [tex]\( x + 3 = 0 \)[/tex], which occurs at [tex]\( x = -3 \)[/tex].
- Therefore, the vertex has an [tex]\( x \)[/tex]-value of [tex]\( -3 \)[/tex], not 0.
Based on this analysis, the functions that have a vertex with an [tex]$x$[/tex]-value of 0 are:
- [tex]\( f(x) = |x| \)[/tex]
- [tex]\( f(x) = |x| + 3 \)[/tex]
- [tex]\( f(x) = |x| - 6 \)[/tex]
Thus, the correct options are:
1. [tex]\( f(x) = |x| \)[/tex]
2. [tex]\( f(x) = |x| + 3 \)[/tex]
4. [tex]\( f(x) = |x| - 6 \)[/tex]
So, the indices of the functions are [tex]\([1, 2, 4]\)[/tex].
1. [tex]\( f(x) = |x| \)[/tex]
- The function [tex]\( f(x) = |x| \)[/tex] has its vertex at the point where the expression inside the absolute value is zero, which happens when [tex]\( x = 0 \)[/tex].
- Thus, the vertex is at [tex]\( (0,0) \)[/tex], and the [tex]\( x \)[/tex]-value of the vertex is indeed 0.
2. [tex]\( f(x) = |x| + 3 \)[/tex]
- The function [tex]\( f(x) = |x| + 3 \)[/tex] is a vertical shift of the function [tex]\( f(x) = |x| \)[/tex].
- The vertex of [tex]\( f(x) = |x| \)[/tex] is at [tex]\( (0,0) \)[/tex]. Adding 3 shifts this vertex up by 3 units but does not change the [tex]\( x \)[/tex]-value.
- Therefore, the vertex remains at [tex]\( x = 0 \)[/tex].
3. [tex]\( f(x) = |x+3| \)[/tex]
- The function [tex]\( f(x) = |x+3| \)[/tex] reaches its minimum when the expression inside the absolute value is zero, which occurs at [tex]\( x = -3 \)[/tex].
- Therefore, the vertex has an [tex]\( x \)[/tex]-value of [tex]\( -3 \)[/tex], not 0.
4. [tex]\( f(x) = |x| - 6 \)[/tex]
- The function [tex]\( f(x) = |x| - 6 \)[/tex] is a vertical shift of the function [tex]\( f(x) = |x| \)[/tex].
- The vertex of [tex]\( f(x) = |x| \)[/tex] is at [tex]\( (0,0) \)[/tex]. Subtracting 6 shifts this vertex down by 6 units but does not change the [tex]\( x \)[/tex]-value.
- Therefore, the vertex remains at [tex]\( x = 0 \)[/tex].
5. [tex]\( f(x) = |x+3| - 6 \)[/tex]
- Similar to [tex]\( f(x) = |x+3| \)[/tex], the [tex]\( x \)[/tex]-value of the vertex will be determined by the expression [tex]\( x + 3 = 0 \)[/tex], which occurs at [tex]\( x = -3 \)[/tex].
- Therefore, the vertex has an [tex]\( x \)[/tex]-value of [tex]\( -3 \)[/tex], not 0.
Based on this analysis, the functions that have a vertex with an [tex]$x$[/tex]-value of 0 are:
- [tex]\( f(x) = |x| \)[/tex]
- [tex]\( f(x) = |x| + 3 \)[/tex]
- [tex]\( f(x) = |x| - 6 \)[/tex]
Thus, the correct options are:
1. [tex]\( f(x) = |x| \)[/tex]
2. [tex]\( f(x) = |x| + 3 \)[/tex]
4. [tex]\( f(x) = |x| - 6 \)[/tex]
So, the indices of the functions are [tex]\([1, 2, 4]\)[/tex].
