State the various transformations applied to the base function ƒ(x) = x^2 to obtain a graph of the function g(x) = −2[(x − 1)^2 + 3]. (A) A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units. (B) A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. (C) A reflection about the y-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. (D) A reflection about the y-axis, a vertical stretch by a factor of 2, a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units.

Answer: Option BA reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. Step-by-step explanation:If the graph of the function [tex]g(x)=cf(h-h) +b[/tex]  represents the transformations made to the graph of [tex]y= f(x)[/tex]  then, by definition: If  [tex]0 <c <1[/tex] then the graph is compressed vertically by a factor c. If  [tex]|c| > 1[/tex] then the graph is stretched vertically by a factor c If [tex]c <0[/tex]  then the graph is reflected on the x axis. If [tex]b> 0[/tex] the graph moves vertically  upwards b units. If [tex]b <0[/tex] the graph moves vertically down  b unitsIf [tex]h> 0[/tex] then the graph of f(x) moves horizontally h units to the left If [tex]h <0[/tex] then the graph of f(x) moves horizontally h units to the rightIn this problem we have the function [tex]g(x) = -2((x - 1)^2 + 3)[/tex] and our parent function is [tex]f(x) = x^2[/tex] therefore it is true that [tex]c =-2<0[/tex]  and [tex]b =-6 <0[/tex] and [tex]h=-1<0[/tex] Therefore the graph is reflected on the x axis, stretched vertically by a factor 2. The graph of f(x) moves horizontally 1 units to the right and shift downward of 6 units. The answer is (B) A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units.