In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation y=ax2+bx+c, which was h=−b(2a). Prove this formula using calculus.

To prove the formula \( h = -\frac{b}{2a} \) for the position of the maximum or minimum of a quadratic equation using calculus, follow these steps:

### Step 1: Express the quadratic function
The general form of a quadratic equation is:

\[
y = ax^2 + bx + c
\]

where \( a \), \( b \), and \( c \) are constants.

### Step 2: Find the first derivative
To find the critical points (which include the maximum or minimum points), we need to calculate the first derivative of the function \( y \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{d}{dx}(ax^2 + bx + c)
\]

Applying the power rule to each term:

\[
\frac{dy}{dx} = 2ax + b
\]

### Step 3: Set the first derivative to zero
The maximum or minimum occurs where the derivative is zero. Therefore, set the first derivative equal to zero and solve for \( x \):

\[
2ax + b = 0
\]

\[
2ax = -b
\]

\[
x = -\frac{b}{2a}
\]

### Step 4: Identify the nature of the critical point
To determine whether this critical point is a maximum or a minimum, we can analyze the second derivative of the function.

Find the second derivative:

\[
\frac{d^2y}{dx^2} = \frac{d}{dx}(2ax + b) = 2a
\]

- If \( a > 0 \), the second derivative \( 2a \) is positive, indicating that the function is concave up, and the critical point is a **minimum**.
- If \( a < 0 \), the second derivative \( 2a \) is negative, indicating that the function is concave down, and the critical point is a **maximum**.

### Conclusion
Thus, the \( x \)-coordinate of the vertex (which is the maximum or minimum point) of the quadratic function \( y = ax^2 + bx + c \) is \( h = -\frac{b}{2a} \), as required.

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Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the y-coordinate of the vertex once you know the x-coordinate?
2. What is the significance of the second derivative in determining the nature of critical points?
3. How does the sign of \( a \) affect the shape of the parabola?
4. Can you apply this method to a cubic function? Why or why not?
5. How does completing the square confirm the vertex formula?
6. What are the applications of the vertex formula in real-world problems?
7. How does the vertex formula relate to the axis of symmetry of a parabola?
8. How would you extend this to find the extrema of higher-degree polynomials?

**Tip:** When working with quadratic equations, always remember that the sign of the leading coefficient \( a \) determines whether the parabola opens upwards (minimum) or downwards (maximum).

Learn how to prove the vertex formula h = -b / (2a) for quadratic equations using calculus. This problem covers key concepts in calculus and quadratic equations, including the first derivative test. Suitable for advanced high school students (Grades 11-12).

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