Vertex Form Calculator – Graph Parabolas

What Is Vertex Form?

Vertex form is a way of writing a quadratic function that shows the vertex directly. It is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

For f(x) = 2(x - 3)² + 1, the vertex is (3, 1). The parabola opens up because a is positive. The number 2 makes it narrower than a normal parabola. You can read all of that from the form without any calculation — that is why vertex form is so useful.

How to Convert Standard Form to Vertex Form

Most quadratics start in standard form, f(x) = ax² + bx + c. To get vertex form, complete the square. Take the example f(x) = x² - 4x + 3:

1. Take half of b and square it. Half of -4 is -2, and (-2)² = 4.
2. Add and subtract that number. f(x) = (x² - 4x + 4) - 4 + 3.
3. Write the square. x² - 4x + 4 is (x - 2)². So f(x) = (x - 2)² - 1.

The vertex is (2, -1). You can check it with the vertex formula x = -b / (2a): -(-4) / 2 = 2, and f(2) = -1. Both ways give the same answer. If a is not 1, factor it out first, then complete the square inside the brackets. The graphing quadratic functions calculator draws the same parabola from standard form, so you can compare both forms on one graph.

What Each Part of Vertex Form Does

Every number in f(x) = a(x - h)² + k changes the graph in a simple way:

• a — direction and width. Positive opens up, negative opens down. A larger |a| makes the parabola narrower; a smaller |a| makes it wider.
• h — left and right. The vertex moves to x = h. A positive h moves the parabola right, a negative h moves it left.
• k — up and down. The vertex moves to y = k. A positive k moves the parabola up, a negative k moves it down.

These are transformations. Start with y = x², the basic parabola with vertex (0, 0). Then a stretches it, h slides it sideways, and k slides it up or down. Once you see that, you can sketch any parabola from vertex form in seconds.

Finding Roots and the Axis of Symmetry

The axis of symmetry is the vertical line through the vertex: x = h. For f(x) = (x - 2)² - 1, it is x = 2. The parabola is a mirror image on each side of this line.

To find the roots, set the function equal to zero and solve. For f(x) = (x - 2)² - 1 = 0, we get (x - 2)² = 1, so x - 2 = 1 or x - 2 = -1. The roots are x = 3 and x = 1. Notice that the roots are the same distance from the axis of symmetry: 3 and 1 are both 1 away from 2. That is always true.

If the vertex is above the x-axis and the parabola opens up, there are no real roots. For f(x) = (x - 2)² + 1, the lowest point is y = 1, so the graph never touches the x-axis.

Vertex Form in Real Life

Vertex form is the form engineers use when they need the highest or lowest point of something. A bridge arch is a parabola, and its vertex is the highest point of the arch. A satellite dish is a parabola, and the vertex is where the signal is collected.

Practice With Our Free Tool

Use the vertex form calculator above to check your homework. Enter the function in vertex form, see the parabola, and read off the vertex, axis and roots. Everything is free, with no download and no sign-up.

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