https://r2.erweima.ai/mathbot/image/1726160103657plb0e0u3.jpg

The function given in the image is:

\[
f(x) = -4x + (2 - x)^2
\]

You are asked to develop this expression and make a vertical shift by 2 units upward. Let's go through the steps to simplify and shift the function:

### 1. Expand \( (2 - x)^2 \):
\[
(2 - x)^2 = 2^2 - 2(2)(x) + x^2 = 4 - 4x + x^2
\]

So the function becomes:
\[
f(x) = -4x + (4 - 4x + x^2)
\]

### 2. Combine like terms:
\[
f(x) = -4x + 4 - 4x + x^2
\]
\[
f(x) = x^2 - 8x + 4
\]

### 3. Shift the function by 2 units upward:
To shift the function vertically by 2 units, add 2 to the expression:
\[
f(x) = x^2 - 8x + 4 + 2 = x^2 - 8x + 6
\]

### Final result:
The developed and shifted function is:
\[
f(x) = x^2 - 8x + 6
\]

Let me know if you need further details or have any questions!

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Here are 5 related questions to deepen your understanding:
1. How would you graph the original function \( f(x) = -4x + (2 - x)^2 \)?
2. What is the effect of shifting a function vertically on its graph?
3. How can you factor the final quadratic \( f(x) = x^2 - 8x + 6 \)?
4. What is the vertex form of a quadratic equation, and how does it relate to the function given?
5. Can this quadratic function be solved for its roots using the quadratic formula?

**Tip:** When expanding squared binomials like \( (a - b)^2 \), always use the formula \( a^2 - 2ab + b^2 \).

Develop the function f(x) = -4x + (2 - x)^2 and shift it 2 units upward.

Explore how to expand the quadratic function f(x) = -4x + (2 - x)^2 and perform a vertical shift by 2 units. This problem involves algebraic binomial expansion and function transformations. Suitable for students in Grades 9-11.

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