f(x) = −x^2 − 4x + 2 write in standard form


To rewrite the given quadratic function \( f(x) = -x^2 - 4x + 2 \) in standard form, we need to express it as:

\[
f(x) = a(x - h)^2 + k
\]

where \(a\), \(h\), and \(k\) are constants, and \((h, k)\) represents the vertex of the parabola.

### Step-by-step process:

1. **Factor out the coefficient of the quadratic term** (if necessary):
   In this case, the coefficient of \(x^2\) is already \(-1\), so we proceed without factoring for now.
   
   \[
   f(x) = - (x^2 + 4x) + 2
   \]

2. **Complete the square**:
   To complete the square inside the parentheses, take half of the coefficient of \(x\) (which is \(4\)), square it, and add/subtract it inside the equation.

   Half of \(4\) is \(2\), and \(2^2 = 4\). So, we add and subtract \(4\) inside the parentheses:
   
   \[
   f(x) = - (x^2 + 4x + 4 - 4) + 2
   \]
   
   Simplifying this gives:
   
   \[
   f(x) = - \left[(x + 2)^2 - 4\right] + 2
   \]

3. **Simplify the expression**:
   Distribute the negative sign and simplify:

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

   \[
   f(x) = - (x + 2)^2 + 6
   \]

### Final standard form:

\[
f(x) = - (x + 2)^2 + 6
\]

This is the standard form of the quadratic function. Here:
- \(a = -1\)
- \(h = -2\)
- \(k = 6\)

The vertex of the parabola is at \((-2, 6)\).

Would you like further details or have any questions?

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Here are 5 questions to expand on this concept:

1. How do you find the vertex of a parabola given the standard form?
2. What does the sign of the coefficient \(a\) tell us about the graph of a quadratic function?
3. How does completing the square help in graphing quadratic functions?
4. Can we convert a quadratic function in standard form back to its general form?
5. How do transformations of \(f(x) = x^2\) relate to changes in the quadratic function?

**Tip:** Completing the square is a useful technique not only for rewriting quadratics but also for solving quadratic equations!

f(x) = −x^2 − 4x + 2 write in standard form

Learn how to rewrite the quadratic function f(x) = -x^2 - 4x + 2 in standard form by completing the square. This step-by-step guide explains key algebraic concepts, including the process of completing the square and finding the vertex. Suitable for students in Grades 9-11.

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