Question

"What are the transformations applied to the function f(x)=−1/2​(x+4)2 based on the following descriptions: reflected over the x-axis, wider by a factor of 1/2​, and moved left 4 units? a) Reflected over the x-axis, stretched vertically by a factor of 1/2, and shifted left 4 units.

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

Answer 1

The transformed function f(x) = -½(x + 4)² has undergone a reflection over the x-axis, a vertical stretch by a factor of 2, and shifted left by 4 units. The correct option is a) Reflected over the x-axis, stretched vertically by a factor of 1/2, and shifted left 4 units.

Step-by-step explanation:

To determine which transformation has been applied to the function f(x) = -½(x + 4)², let's analyze each part of the function:

• The negative sign in front of the fraction indicates a reflection over the x-axis.
• The fraction ½ represents a vertical stretch by a factor of 2, which is often mistakenly referred to as a compression by a factor of 2 because the number is less than 1, but in reality it makes the function wider (a true compression would be by a factor greater than 1).
• The term (x + 4) implies a horizontal translation to the left by 4 units because adding a positive number inside the function's argument moves it in the opposite direction.

Considering these points, the correct description of the transformations applied to the original function is a reflection over the x-axis, a vertical stretch (making the function wider) by a factor of 2, and a shift to the left by 4 units. Hence, the correct option is a) Reflected over the x-axis, stretched vertically by a factor of 1/2, and shifted left 4 units.