Answer :
To analyze the function [tex]\( f(x) = -6|x + 5| - 2 \)[/tex] and determine which statement about the function is true, we will carefully examine each component of the function step-by-step.
1. Understanding the basic form of the function:
The function [tex]\( f(x) = -6|x + 5| - 2 \)[/tex] is a transformation of the parent function [tex]\( g(x) = |x| \)[/tex].
2. Vertical transformations:
- The term [tex]\(-6|x + 5|\)[/tex] indicates that we multiply the absolute value of [tex]\((x + 5)\)[/tex] by [tex]\(-6\)[/tex]. This negative coefficient indicates that the graph will be reflected across the x-axis, meaning it opens downward.
- The factor of [tex]\(-6\)[/tex] also indicates a vertical stretch. This means the graph is stretched vertically by a factor of 6 compared to the parent function [tex]\(|x|\)[/tex].
3. Horizontal transformations:
- The [tex]\(x+5\)[/tex] inside the absolute value modifies the graph horizontally. This means the graph is shifted to the left by 5 units.
4. Vertical shift:
- The [tex]\(-2\)[/tex] at the end of the function indicates a vertical shift downward by 2 units.
Now, let's analyze each statement:
1. The graph of [tex]\( f(x) \)[/tex] is a horizontal compression of the graph of the parent function.
- A horizontal compression occurs if there is a factor inside the absolute value that stretches or compresses the graph horizontally, which is not the case here. Thus, this statement is false.
2. The graph of [tex]\( f(x) \)[/tex] is a horizontal stretch of the graph of the parent function.
- Similarly, a horizontal stretch would be indicated by a factor that causes the graph to widen or extend horizontally, which is again not the case here. The horizontal shift due to [tex]\(x + 5\)[/tex] is a translation, not a stretch. Thus, this statement is false.
3. The graph of [tex]\( f(x) \)[/tex] opens upward.
- Due to the negative coefficient [tex]\(-6\)[/tex] in front of the absolute value, the graph opens downward. Therefore, this statement is false.
4. The graph of [tex]\( f(x) \)[/tex] opens to the right.
- The notion of a graph "opening to the right" is not appropriate in this context. The absolute value function inherently does not "open to the right," as it deals with vertical stretching and positions along the x-axis. Therefore, this statement is false.
Given the analysis and the nature of the function, it is clear that:
None of the statements provided is true.
1. Understanding the basic form of the function:
The function [tex]\( f(x) = -6|x + 5| - 2 \)[/tex] is a transformation of the parent function [tex]\( g(x) = |x| \)[/tex].
2. Vertical transformations:
- The term [tex]\(-6|x + 5|\)[/tex] indicates that we multiply the absolute value of [tex]\((x + 5)\)[/tex] by [tex]\(-6\)[/tex]. This negative coefficient indicates that the graph will be reflected across the x-axis, meaning it opens downward.
- The factor of [tex]\(-6\)[/tex] also indicates a vertical stretch. This means the graph is stretched vertically by a factor of 6 compared to the parent function [tex]\(|x|\)[/tex].
3. Horizontal transformations:
- The [tex]\(x+5\)[/tex] inside the absolute value modifies the graph horizontally. This means the graph is shifted to the left by 5 units.
4. Vertical shift:
- The [tex]\(-2\)[/tex] at the end of the function indicates a vertical shift downward by 2 units.
Now, let's analyze each statement:
1. The graph of [tex]\( f(x) \)[/tex] is a horizontal compression of the graph of the parent function.
- A horizontal compression occurs if there is a factor inside the absolute value that stretches or compresses the graph horizontally, which is not the case here. Thus, this statement is false.
2. The graph of [tex]\( f(x) \)[/tex] is a horizontal stretch of the graph of the parent function.
- Similarly, a horizontal stretch would be indicated by a factor that causes the graph to widen or extend horizontally, which is again not the case here. The horizontal shift due to [tex]\(x + 5\)[/tex] is a translation, not a stretch. Thus, this statement is false.
3. The graph of [tex]\( f(x) \)[/tex] opens upward.
- Due to the negative coefficient [tex]\(-6\)[/tex] in front of the absolute value, the graph opens downward. Therefore, this statement is false.
4. The graph of [tex]\( f(x) \)[/tex] opens to the right.
- The notion of a graph "opening to the right" is not appropriate in this context. The absolute value function inherently does not "open to the right," as it deals with vertical stretching and positions along the x-axis. Therefore, this statement is false.
Given the analysis and the nature of the function, it is clear that:
None of the statements provided is true.
