Answer :
Let's analyze the provided function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] and the given statements to determine which, if any, is true.
### Step 1: Determine the Vertex:
The vertex form of an absolute value function is [tex]\( f(x) = a |x - h| + k \)[/tex], where [tex]\((h, k)\)[/tex] represents the vertex of the graph.
- For the given function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex], we can identify [tex]\( h \)[/tex] and [tex]\( k \)[/tex] from the expression inside and outside the absolute value.
- Comparing [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] with the standard form [tex]\( f(x) = a |x - h| + k \)[/tex], we see that [tex]\( h = -4 \)[/tex] and [tex]\( k = -6 \)[/tex].
Thus, the vertex of the function is [tex]\((-4, -6)\)[/tex].
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] has a vertex of [tex]\((-4,6)\)[/tex]:
This statement is false because the vertex is actually [tex]\((-4, -6)\)[/tex].
### Step 2: Check for Horizontal Stretch:
A horizontal stretch affects the [tex]\( x \)[/tex]-values of the function, expanding or compressing the graph horizontally. The function [tex]\( a |x - h| + k \)[/tex] with [tex]\( a \neq 1 \)[/tex] affects the vertical stretch/compression and reflection.
- [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] has a factor of [tex]\(-\frac{2}{3}\)[/tex]. This means a vertical compression by a factor of [tex]\( \frac{2}{3} \)[/tex] and a reflection across the x-axis, not a horizontal stretch or compression.
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] is a horizontal stretch of the graph of the parent function:
This statement is false. The transformation is a vertical compression and a downward reflection, not a horizontal stretch.
### Step 3: Determine the Direction of Opening:
The sign of the coefficient [tex]\( a \)[/tex] in [tex]\( f(x) = a |x - h| + k \)[/tex] determines if the graph opens upward or downward.
- Here, [tex]\( a = -\frac{2}{3} \)[/tex], which is negative.
Thus, the graph opens downward.
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] opens upward:
This statement is false. The graph opens downward due to the negative coefficient.
### Step 4: Determine the Domain:
The domain of the absolute value function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] is all real numbers.
- Absolute value functions naturally have a domain of all real numbers unless explicitly restricted.
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] has a domain of [tex]\( x \leq -6 \)[/tex]:
This statement is false. The domain is all real numbers.
### Conclusion:
After analyzing all the given statements and the characteristics of the function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex], it turns out that none of the statements provided are true.
Therefore, the overall conclusion is that there is no true statement out of the given options.
### Step 1: Determine the Vertex:
The vertex form of an absolute value function is [tex]\( f(x) = a |x - h| + k \)[/tex], where [tex]\((h, k)\)[/tex] represents the vertex of the graph.
- For the given function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex], we can identify [tex]\( h \)[/tex] and [tex]\( k \)[/tex] from the expression inside and outside the absolute value.
- Comparing [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] with the standard form [tex]\( f(x) = a |x - h| + k \)[/tex], we see that [tex]\( h = -4 \)[/tex] and [tex]\( k = -6 \)[/tex].
Thus, the vertex of the function is [tex]\((-4, -6)\)[/tex].
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] has a vertex of [tex]\((-4,6)\)[/tex]:
This statement is false because the vertex is actually [tex]\((-4, -6)\)[/tex].
### Step 2: Check for Horizontal Stretch:
A horizontal stretch affects the [tex]\( x \)[/tex]-values of the function, expanding or compressing the graph horizontally. The function [tex]\( a |x - h| + k \)[/tex] with [tex]\( a \neq 1 \)[/tex] affects the vertical stretch/compression and reflection.
- [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] has a factor of [tex]\(-\frac{2}{3}\)[/tex]. This means a vertical compression by a factor of [tex]\( \frac{2}{3} \)[/tex] and a reflection across the x-axis, not a horizontal stretch or compression.
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] is a horizontal stretch of the graph of the parent function:
This statement is false. The transformation is a vertical compression and a downward reflection, not a horizontal stretch.
### Step 3: Determine the Direction of Opening:
The sign of the coefficient [tex]\( a \)[/tex] in [tex]\( f(x) = a |x - h| + k \)[/tex] determines if the graph opens upward or downward.
- Here, [tex]\( a = -\frac{2}{3} \)[/tex], which is negative.
Thus, the graph opens downward.
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] opens upward:
This statement is false. The graph opens downward due to the negative coefficient.
### Step 4: Determine the Domain:
The domain of the absolute value function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex] is all real numbers.
- Absolute value functions naturally have a domain of all real numbers unless explicitly restricted.
#### Statement Analysis:
- The graph of [tex]\( f(x) \)[/tex] has a domain of [tex]\( x \leq -6 \)[/tex]:
This statement is false. The domain is all real numbers.
### Conclusion:
After analyzing all the given statements and the characteristics of the function [tex]\( f(x) = -\frac{2}{3} |x + 4| - 6 \)[/tex], it turns out that none of the statements provided are true.
Therefore, the overall conclusion is that there is no true statement out of the given options.
