Answer :
To find the axis of symmetry for the function [tex]\( f(x) = 4|x + 8| - 5 \)[/tex], we need to understand the general form of an absolute value function.
The standard form of an absolute value function is [tex]\( f(x) = a|x - h| + k \)[/tex], where:
- [tex]\( a \)[/tex] is a constant that affects the "steepness" or width of the graph,
- [tex]\( h \)[/tex] is the horizontal shift, and
- [tex]\( k \)[/tex] is the vertical shift.
In this function, the center (or vertex) of the V-shape occurs at [tex]\( x = h \)[/tex]. The axis of symmetry is a vertical line that passes through the vertex, so it is given by the equation [tex]\( x = h \)[/tex].
For the function [tex]\( f(x) = 4|x + 8| - 5 \)[/tex]:
- The expression inside the absolute value is [tex]\( x + 8 \)[/tex].
- This can be rewritten in the form [tex]\( |x - (-8)| \)[/tex], indicating a shift of 8 units to the left.
Thus, the value of [tex]\( h \)[/tex] is [tex]\(-8\)[/tex].
Therefore, the axis of symmetry for the function is [tex]\( x = -8 \)[/tex].
The correct answer is not listed in the options given. However, based on the understanding of how to find the axis of symmetry, we have determined it to be [tex]\( x = -8 \)[/tex].
The standard form of an absolute value function is [tex]\( f(x) = a|x - h| + k \)[/tex], where:
- [tex]\( a \)[/tex] is a constant that affects the "steepness" or width of the graph,
- [tex]\( h \)[/tex] is the horizontal shift, and
- [tex]\( k \)[/tex] is the vertical shift.
In this function, the center (or vertex) of the V-shape occurs at [tex]\( x = h \)[/tex]. The axis of symmetry is a vertical line that passes through the vertex, so it is given by the equation [tex]\( x = h \)[/tex].
For the function [tex]\( f(x) = 4|x + 8| - 5 \)[/tex]:
- The expression inside the absolute value is [tex]\( x + 8 \)[/tex].
- This can be rewritten in the form [tex]\( |x - (-8)| \)[/tex], indicating a shift of 8 units to the left.
Thus, the value of [tex]\( h \)[/tex] is [tex]\(-8\)[/tex].
Therefore, the axis of symmetry for the function is [tex]\( x = -8 \)[/tex].
The correct answer is not listed in the options given. However, based on the understanding of how to find the axis of symmetry, we have determined it to be [tex]\( x = -8 \)[/tex].