Question

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 1

Answer: 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.

Explanation:

If the graph of the function
`g(x)=cf(h-h) +b` represents the transformations made to the graph of
`y= f(x)` then, by definition:

If
`0 <c <1` then the graph is compressed vertically by a factor c.

If
`|c| > 1` then the graph is stretched vertically by a factor c

If
`c <0` then the graph is reflected on the x axis.

If
`b> 0` the graph moves vertically upwards b units.

If
`b <0` the graph moves vertically down b units

If
`h> 0` then the graph of f(x) moves horizontally h units to the left

If
`h <0` then the graph of f(x) moves horizontally h units to the right

In this problem we have the function
`g(x) = -2((x - 1)^2 + 3)` and our parent function is
`f(x) = x^2`

therefore it is true that
`c =-2<0` and
`b =-6 <0` and
`h=-1<0`

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.