Question

For the given function

�
(
�
)
=
2
−
1
2
�
+
2
+
1
f(x)=2
−
2
1
​
x+2
​
+1, identify all (5) transformations that have been applied. Remember to factor first.

A) Vertical stretch by 2, reflection over the x-axis, horizontal compression by 2, right shift of 2 units, vertical shift up by 1 unit

B) Horizontal stretch by 2, reflection over the y-axis, vertical compression by 2, left shift of 2 units, vertical shift up by 1 unit

C) Vertical stretch by 2, reflection over the x-axis, horizontal compression by 2, left shift of 2 units, vertical shift up by 1 unit

D) Horizontal stretch by 2, reflection over the y-axis, vertical compression by 2, right shift of 2 units, vertical shift up by 1 unit

Answer 1

The correct option is A) Vertical stretch by 2, reflection over the x-axis, horizontal compression by 2, right shift of 2 units, vertical shift up by 1 unit.

So, the correct option is A.

Step-by-step explanation:

The given function can be expressed as
`\(f(x) = 2 - (1)/(2)x + 2 + 1\).`Let's break down the transformations:

1. Vertical Stretch by 2: The coefficient of the \(x\) term is multiplied by 2, causing a vertical stretch.

2. Reflection over the x-axis: The negative sign in front of the
`\((1)/(2)x\)` term indicates a reflection over the x-axis.

3. Horizontal Compression by 2: The reciprocal of the coefficient of the \(x\) term is taken, leading to a horizontal compression.

4. Right Shift of 2 Units: The constant term (+2) inside the function implies a right shift of 2 units.

5. Vertical Shift Up by 1 Unit: The constant term (+1) outside the function suggests a vertical shift upwards by 1 unit.

Each transformation is determined by specific characteristics in the function's expression, and these alterations combine to create the overall transformation of the given function.

In conclusion, by identifying and understanding the role of each component in the function, we can accurately determine the sequence of transformations applied to it. The final expression reflects the cumulative effect of these transformations on the original function, resulting in the correct choice, option A, as the accurate description of the transformations applied.

So, the correct option is A.