Vertex Form Calculator – Guide & Formulas
Convert quadratic equations to vertex form y = a(x-h)² + k. Find the vertex, axis of symmetry, and direction of opening.
Convert a quadratic equation from standard form (ax² + bx + c) to vertex form (a(x-h)² + k). Find the vertex coordinates, axis of symmetry, and direction of opening.
Key Takeaway
Use the free Vertex Form Calculator to convert quadratic equations to vertex form y = a(x-h)² + k. find the vertex, axis of symmetry, and direction of opening. Get instant results with step-by-step explanations.
How to Use the Vertex Form Calculator
- Enter the coefficients a, b, and c from the standard form ax² + bx + c.
- Click "Calculate" to convert to vertex form.
- Review the vertex (h, k), axis of symmetry, and direction of opening.
- The step-by-step completion of the square is shown.
The Formula
Variable Definitions
- a: Leading coefficient (determines width and direction of opening)
- h: x-coordinate of the vertex (also axis of symmetry)
- k: y-coordinate of the vertex (maximum or minimum value)
- b: Linear coefficient
- c: Constant term (y-intercept)
Example: Convert y = 2x² - 8x + 5 to Vertex Form
Convert the quadratic y = 2x² - 8x + 5 to vertex form and find the vertex.
- Step 1: Identify a = 2, b = -8, c = 5.
- Step 2: Calculate h = -b/(2a) = -(-8)/(2×2) = 8/4 = 2.
- Step 3: Calculate k = c - b²/(4a) = 5 - (-8)²/(4×2) = 5 - 64/8 = 5 - 8 = -3.
- Step 4: Vertex form: y = 2(x - 2)² - 3.
- Step 5: Vertex is (2, -3), axis of symmetry is x = 2, parabola opens upward.
Frequently Asked Questions
What is vertex form?
Vertex form is y = a(x-h)² + k where (h,k) is the vertex of the parabola. It makes it easy to identify the vertex, axis of symmetry, and direction of opening.
How do I convert standard form to vertex form?
Use completing the square: factor out a from the x terms, add and subtract (b/2a)² inside the parentheses, then simplify to get the vertex form.
What is the vertex of a parabola?
The vertex is the highest or lowest point of a parabola. For y = a(x-h)² + k, the vertex is at (h, k). If a > 0, it is a minimum; if a < 0, it is a maximum.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = h that passes through the vertex. It divides the parabola into two mirror-image halves.
What does the value of a tell us?
The value of a determines the width and direction. If a > 0, the parabola opens upward. If a < 0, it opens downward. Larger |a| means a narrower parabola.
How do I find the y-intercept from vertex form?
Set x = 0: y = a(0-h)² + k = ah² + k. This gives the y-intercept point (0, ah² + k).
What is completing the square?
Completing the square is an algebraic technique to rewrite a quadratic as a perfect square trinomial plus a constant, which gives the vertex form.
Can I convert vertex form back to standard form?
Yes, expand the vertex form: y = a(x-h)² + k = ax² - 2ahx + ah² + k. This gives the standard form coefficients.
What is the relationship between vertex form and factored form?
Vertex form highlights the vertex, while factored form y = a(x-r₁)(x-r₂) highlights the roots (x-intercepts). Both are equivalent representations.
How is vertex form used in real life?
Vertex form is used in projectile motion (finding maximum height), optimization problems (finding minimum/maximum), and modeling parabolic trajectories.