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.

Option 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.

Step-by-step explanation:

If the graph of the function  represents the transformations made to the graph of  then, by definition:

If   then the graph is compressed vertically by a factor c.

If   then the graph is stretched vertically by a factor c

If  then the graph is reflected on the x axis.

If the graph moves vertically  upwards b units.

If the graph moves vertically down  b units

If then the graph of f(x) moves horizontally h units to the left

If then the graph of f(x) moves horizontally h units to the right

In this problem we have the function and our parent function is

therefore it is true that  and and

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.