use transformations to graph the function and state the domain and range. y = -\\frac{1}{6}|x| + 15 use the…

Explanation:

Step1: Identify the parent - function

The parent - function of (y =-\frac{1}{6}|x| + 15) is (y = |x|). The graph of (y = |x|) is a V - shaped graph with the vertex at the origin ((0,0)) and it opens upwards.

Step2: Analyze the vertical stretch and reflection

The coefficient (-\frac{1}{6}) in front of (|x|) causes a vertical compression by a factor of (\frac{1}{6}) and a reflection across the (x) - axis.

Step3: Analyze the vertical translation

The (+ 15) at the end of the function causes a vertical shift upwards by 15 units. The vertex of the new function (y=-\frac{1}{6}|x| + 15) is at the point ((0,15)).

Step4: Determine the domain

The domain of the absolute - value function (y =-\frac{1}{6}|x|+15) is all real numbers since we can substitute any real number for (x) into the function. In interval notation, the domain is ((-\infty,\infty)).

Step5: Determine the range

Since the graph is reflected across the (x) - axis and shifted up 15 units and has a maximum value at the vertex. The range is (y\leq15). In interval notation, the range is ((-\infty,15]).

Answer:

Domain: ((-\infty,\infty)); Range: ((-\infty,15])