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).