Let's break down the behavior of the function [tex]\( f(x) = -6|x+5| - 2 \)[/tex] and analyze which statement is true.
1. Vertical Reflection:
The function [tex]\( f(x) \)[/tex] includes a term [tex]\(-6|x+5|\)[/tex]. The negative sign in front of the absolute value indicates a vertical reflection. This means the graph of the function is flipped over the x-axis compared to the parent absolute value function [tex]\( |x| \)[/tex].
2. Opening Direction:
Because of the vertical reflection indicated by the negative sign, the graph opens downward, not upward. Thus, the statement "The graph of [tex]\(f(x)\)[/tex] opens upward" is false.
3. Vertical Stretch:
The coefficient -6 not only reflects the graph but also stretches it vertically. This is not a horizontal stretch or compression but a vertical transformation.
4. Horizontal Shift:
The function includes [tex]\( |x+5| \)[/tex], which means the entire graph of the absolute value function [tex]\( |x| \)[/tex] is shifted horizontally to the left by 5 units (since +5 inside the absolute value moves the graph in the negative x-direction).
5. Vertical Shift:
The constant term [tex]\(-2\)[/tex] shifts the entire graph downward by 2 units. This is a vertical shift rather than anything that would affect the horizontal direction of the graph.
Given these points, we can analyze the statements provided:
- The graph of [tex]\(f(x)\)[/tex] is a horizontal compression of the graph of the parent function. This is false, as the transformation is a vertical stretch and not a horizontal compression.
- The graph of [tex]\(f(x)\)[/tex] is a horizontal stretch of the graph of the parent function. This is false for the same reason; the graph undergoes a vertical stretch and reflection, not a horizontal stretch.
- The graph of [tex]\(f(x)\)[/tex] opens upward. This is false because the negative sign causes the graph to open downward, not upward.
- The graph of [tex]\(f(x)\)[/tex] opens to the right. This is also false, as the term [tex]\( |x+5| \)[/tex] shifts the graph horizontally but does not affect the direction in which it opens, which is downward due to the reflection.
Based on this, none of the statements provided are entirely accurate regarding the graph's transformations correctly.
