which statements about the graph of the function ( f(x)=-x^{2}-4x + 2 ) are true? select three options. the…

Explanation:

Step1: Find the vertex of the parabola

For a quadratic function (y = ax^{2}+bx + c), the (x) - coordinate of the vertex is (x=-\frac{b}{2a}). Given (f(x)=-x^{2}-4x + 2), where (a=-1), (b = - 4), (c = 2). (x=-\frac{-4}{2\times(-1)}=-2). Substitute (x = - 2) into the function: (y=-(-2)^{2}-4\times(-2)+2=-4 + 8+2=6). The vertex form of the function is (y=-(x + 2)^{2}+6).

Step2: Analyze the domain and range

The domain of a quadratic function (y = ax^{2}+bx + c) is all real numbers, (x\in(-\infty,\infty)), so the statement "The domain is ({x|x\leq - 2})" is false. Since (a=-1<0), the parabola opens downwards. The maximum value of the function is (y = 6), so the range is ({y|y\leq6}).

Step3: Analyze the increasing and decreasing intervals

For a parabola (y = a(x - h)^{2}+k) ((a<0)) with vertex ((h,k)), the function is increasing on the interval ((-\infty,h)) and decreasing on the interval ((h,\infty)). Here (h=-2), so the function is increasing on ((-\infty,-2)) and decreasing on ((-2,\infty)). The statement "The function is decreasing over the interval ((-4,\infty))" is false.

Step4: Find the (y) - intercept

To find the (y) - intercept, set (x = 0). Then (y=-0^{2}-4\times0+2=2>0). So the function has a positive (y) - intercept.

Answer:

B. The range is ({y|y\leq6}), C. The function is increasing over the interval ((-\infty,-2)), E. The function has a positive (y) - intercept.