Question

1. how does the transformation ( f(x) = -2(x - 1)^2 + 7 ) change the parent function ( f(x) = x^2 )?

a. reflection over the x - axis, vertical compression by 2, shift left by 1, shift down by 7.
b. reflection over the x - axis, vertical stretch by 2, shift right by 1, shift up by 7.
c. reflection over the y - axis, vertical compression by 2, shift right by 1, shift up by 7.
d. reflection over the y - axis, vertical stretch by 2, shift left by 1, shift up by 7.

Explanation:

Step1: Identify reflection factor

The coefficient of the squared term is $-2$. The negative sign means reflection over the x-axis.

Step2: Identify vertical stretch factor

The absolute value of the coefficient $|-2|=2$, so vertical stretch by 2.

Step3: Identify horizontal shift

The term inside the square is $(x-1)$, which means shift right by 1.

Step4: Identify vertical shift

The constant term at the end is $+7$, which means shift up by 7.

Answer:

b. Reflection over the x-axis, vertical stretch by 2, shift right by 1, shift up by 7.