To analyze which statement is true about the function [tex]\( f(x) + 2 = \frac{1}{6} |x - 3| \)[/tex], we first need to rewrite it in a more familiar form. Let's isolate [tex]\( f(x) \)[/tex] in the given equation:
[tex]\[
f(x) = \frac{1}{6} |x - 3| - 2
\][/tex]
Now that we have the function [tex]\( f(x) \)[/tex], we can analyze each statement one by one.
1. The graph of [tex]\( f(x) \)[/tex] has a vertex of [tex]\( (-3, 2) \)[/tex].
The standard form of an absolute value function is [tex]\( f(x) = a |x - h| + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the graph.
In our function, [tex]\( f(x) = \frac{1}{6} |x - 3| - 2 \)[/tex], we can see that [tex]\( h = 3 \)[/tex] and [tex]\( k = -2 \)[/tex].
Thus, the vertex of the graph is [tex]\( (3, -2) \)[/tex], not [tex]\( (-3, 2) \)[/tex].
So, this statement is false.
2. The graph of [tex]\( f(x) \)[/tex] is a horizontal compression of the graph of the parent function.
A coefficient inside the absolute value affects horizontal stretching or compression. In this case, we have a coefficient outside the absolute value, [tex]\(\frac{1}{6}\)[/tex], which affects vertical compression or stretching.
Since [tex]\( \frac{1}{6} \)[/tex] is outside the absolute value, it describes a vertical compression, not a horizontal one.
Thus, this statement is false.
3. The graph of [tex]\( f(x) \)[/tex] opens downward.
The direction of the opening of an absolute value function is determined by the coefficient [tex]\( a \)[/tex] in [tex]\( f(x) = a|x - h| + k \)[/tex].
Here the coefficient is [tex]\(\frac{1}{6}\)[/tex], which is positive.
Therefore, the graph opens upward.
So, this statement is false.
4. The graph of [tex]\( f(x) \)[/tex] has the range of [tex]\( f(x) \geq -2 \)[/tex].
We analyze the range by looking at the minimum value of [tex]\( f(x) \)[/tex].
The term [tex]\( |x - 3| \)[/tex] is always non-negative ([tex]\( \geq 0 \)[/tex]). The least value of [tex]\( \frac{1}{6} |x - 3| \)[/tex] is [tex]\( 0 \)[/tex], which occurs when [tex]\( x = 3 \)[/tex].
At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = \frac{1}{6}(0) - 2 = -2 \)[/tex].
Since the absolute value term [tex]\( \frac{1}{6} |x - 3| \)[/tex] is always non-negative, any value added to [tex]\( -2 \)[/tex] will be [tex]\( \geq -2 \)[/tex].
Thus, the range of [tex]\( f(x) \)[/tex] is indeed [tex]\( f(x) \geq -2 \)[/tex].
This statement is true.
In conclusion, the correct statement is:
The graph of [tex]\( f(x) \)[/tex] has a range of [tex]\( f(x) \geq -2 \)[/tex].
