Question

Write a rule a g described by the transformation of f(x) = x², then identify the vertex.

Vertical compression of and reflection in the x-axis, followed by a translation 2 units down.
MULTIPLE CHOICE
A.) g(x) = (x - 2)²; (2,0)
x² – 2; (0, -2)
B.) g(x) =
C.)
g(x)=x²-2; (0, 2)
D.) g(x) = -4x² - 2; (0, -2)

Answer 1

The correct transformation of the function f(x) = x², with a vertical compression, reflection in the x-axis, and translation 2 units down, is g(x) = -x² - 2. The vertex of this transformed function is (0, -2).

Step-by-step explanation:

To find the rule of g(x) described by the transformation of f(x) = x², we need to apply the transformations one by one.

1. A vertical compression would change the coefficient of x². However, since no specific factor of compression is given, we can't determine the coefficient.
2. A reflection in the x-axis will change the sign of our function, so f(x) becomes -x².
3. Finally, a translation 2 units down will subtract 2 from our function, resulting in g(x) = -x² - 2.

The vertex of the parabola given by g(x) can be found by setting the derivative of g(x) to zero. However, since this is a vertical transformation of the original parabola f(x) = x² whose vertex is at (0, 0), the vertex of g(x) will be at (0, -2).

Therefore, the correct answer is: D.) g(x) = -x² - 2; Vertex: (0, -2)