Guide
How to use Vertex Form Calculator
Start with the working tool above. The notes below explain the inputs, method, limitations and checks that matter for this specific task.
- 1. Write your quadratic in standard form: y = ax^2 + bx + c.
- 2. Enter coefficients a, b and c, keeping their signs.
- 3. Press Calculate for the vertex, vertex form, axis of symmetry and y-intercept.
- 4. Check the sign of a: positive opens upward (minimum), negative downward (maximum).
- 5. Sketch the graph: plot (h, k), then the y-intercept, and mirror it across the axis.
How it works
The vertex sits on the axis of symmetry, x = -b / (2a), so h = -b / (2a). Substituting back gives k = c - b^2 / (4a), the y-value at the vertex. Vertex form is then y = a(x - h)^2 + k, with the same a as standard form. The y-intercept is simply c, y's value at x = 0.
Worked examples
Example 1: a = 1, b = -4, c = 5
h = -b / (2a) = -(-4) / 2 = 2. k = c - b^2 / (4a) = 5 - 16 / 4 = 5 - 4 = 1. Vertex: (2, 1). Vertex form: y = (x - 2)^2 + 1. Axis of symmetry: x = 2. Y-intercept: 5. Positive a opens upward, so (2, 1) is the minimum.
Example 2: a = -2, b = 8, c = 3 (opens downward)
h = -b / (2a) = -8 / -4 = 2. k = c - b^2 / (4a) = 3 - 64 / -8 = 3 + 8 = 11. Vertex: (2, 11). Vertex form: y = -2(x - 2)^2 + 11. Axis: x = 2. Y-intercept: 3. Negative a opens downward, so (2, 11) is the maximum, the highest value the function reaches.
When this tool is useful
- Finding a projectile's maximum height or a model's minimum cost point.
- Sketching parabolas without completing the square by hand.
- Checking vertex form conversions on homework.
- Locating the optimum in business models, such as the price that maximizes revenue.
Reading Vertex Form
Vertex form is a transformation story. Starting from y = x^2, a stretches the graph vertically (and flips it if negative), h shifts it left or right, and k shifts it up or down. Because the shifts show in the equation, vertex form is the fastest way to sketch a parabola.
Watch the horizontal shift, the most common slip here: y = (x - 2)^2 sits at x = 2, and y = (x + 2)^2 at x = -2. The sign inside the brackets is the opposite of the vertex's x-coordinate.
Minimum and Maximum in Real Problems
The vertex answers optimization questions. A ball's height follows a downward parabola whose vertex is the peak height and when it occurs. A downward-curving profit model has its best price at the vertex; an upward-curving cost model has its cheapest point there.
In each case h gives the input producing the optimum and k gives the optimal value itself, which is why both numbers matter.
Common mistakes to avoid
- Flipping the sign of h. The form is (x - h), so a vertex at x = -3 is written (x + 3)^2, not (x - 3)^2.
- Computing h as b / (2a), forgetting the minus sign in -b / (2a).
- Changing a during conversion; the a in vertex form is exactly the standard-form a.
- Reporting c as the vertex y-value; c is the y-intercept, k is the vertex's y-coordinate.
- Calling the vertex a maximum when a is positive; positive a opens upward, so it is a minimum.
Frequently asked questions
How do I convert standard form to vertex form?
Compute h = -b / (2a) and k = c - b^2 / (4a), then write y = a(x - h)^2 + k with the same a. For x^2 - 4x + 5: h = 2, k = 1, so y = (x - 2)^2 + 1.
What do h and k tell me about the graph?
The vertex is the point (h, k): h is its horizontal position, k its height. Against y = x^2, the graph shifts right by h and up by k, turning at the vertex.
Why is it (x - h) and not (x + h)?
Because the squared term must be zero at the vertex: when x = h, (x - h) is zero and y = k. So positive h shifts right, and negative h, say -3, gives (x + 3), shifting left.
What is the axis of symmetry?
It is the vertical line x = h splitting the parabola into mirror halves. Points match across it at equal heights, so the y-intercept gives a second point free by reflection.
Is the vertex a maximum or a minimum?
Positive a opens upward: the vertex is the minimum. Negative a opens downward: the vertex is the maximum. Larger |a| narrows the parabola; smaller widens it.
What if a equals 0?
Then it is not a quadratic but a straight line, y = bx + c, with no vertex. The calculator requires nonzero a: no x^2 term, no parabola.
Bottom line
Two formulas, h = -b / (2a) and k = c - b^2 / (4a), unlock the graph: the turning point, the axis, and whether it is a maximum or a minimum. Run your coefficients through the calculator, then sanity-check the signs before sketching.
If the question changes shape, the closest neighbours are Sector Area Calculator and Angle of Elevation Calculator — or browse everything under Math Calculators.