Question

"What transformations have been applied to the function f(x)=−1/2​(x−4)2 based on the following instructions: reflected over the x-axis, wider by a factor of 1/2​, and moved right 4 units?" a) Reflected over the y-axis, compressed vertically by a factor of 2, and shifted right 4 units.

b) Reflected over the x-axis, stretched vertically by a factor of 1/2, and shifted left 4 units.
c) Reflected over the y-axis, stretched vertically by a factor of 1/2, and shifted right 4 units.
d) Reflected over the x-axis, compressed vertically by a factor of 2, and shifted left 4 units.

Answer 1

The function f(x) = -1/2(x-4)^2 is reflected over the x-axis, widened by a factor of 1/2, and moved right 4 units to become f(x) = 1/4(x-4)^2.

Step-by-step explanation:

The function f(x) = -1/2(x-4)^2 undergoes three transformations: reflection over the x-axis, widening by a factor of 1/2, and a rightward translation by 4 units.

1. When a function is reflected over the x-axis, the sign of the function is flipped, so the negative sign in the equation stays the same and becomes positive.

2. When a function is widened by a factor of 1/2, the coefficient in front of the x-term changes. In this case, -1/2 becomes -1/4.

3. When a function is moved right 4 units, the x-term in the equation is replaced by (x - 4).

Putting it all together, the transformed function is f(x) = 1/4(x-4)^2.