1. What is the Vertex Formula?

The Vertex Formula calculates the x-coordinate of the vertex of a parabola given by the quadratic equation \( f(x) = ax^2 + bx + c \). The vertex represents either the maximum or minimum point of the parabola.

2. How Does the Calculator Work?

The calculator uses the Vertex Formula:

Where:

• \( a \) — Coefficient of \( x^2 \) term
• \( b \) — Coefficient of \( x \) term

Explanation: The formula is derived from completing the square of the general quadratic equation. The vertex's y-coordinate can be found by plugging h back into the original equation.

3. Importance of Vertex Calculation

Details: Finding the vertex is essential in optimization problems, physics (projectile motion), economics (profit maximization), and graphing quadratic functions.

4. Using the Calculator

Tips: Enter coefficients a and b from your quadratic equation. Coefficient a cannot be zero (as that would make it a linear equation).

5. Frequently Asked Questions (FAQ)

Q1: What if a = 0?
A: The equation becomes linear (not quadratic), and the vertex formula doesn't apply. The calculator will show an error if a=0 is entered.

Q2: How do I find the y-coordinate of the vertex?
A: Plug the h value back into your original equation: \( k = f(h) = a(h)^2 + b(h) + c \).

Q3: What does the vertex represent?
A: It's the maximum point if a < 0 (parabola opens downward) or minimum point if a > 0 (parabola opens upward).

Q4: Can this be used for any quadratic equation?
A: Yes, as long as it's in the standard form \( ax^2 + bx + c \).

Q5: How is this different from the quadratic formula?
A: The quadratic formula finds roots (x-intercepts), while the vertex formula finds the turning point of the parabola.