Vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. The value of a controls whether the parabola opens up or down and how narrow or wide it is. Vertex form is especially useful for graphing and optimization problems.

How do you convert standard form to vertex form?

For f(x)=ax²+bx+c, calculate h = -b/(2a), then calculate k = f(h). After that, write the function as f(x)=a(x-h)²+k. You can also convert by completing the square, which shows the algebraic structure behind the formula.

What is the vertex of a quadratic?

The vertex is the turning point of the parabola. If the parabola opens upward, the vertex is the minimum point. If the parabola opens downward, the vertex is the maximum point. In vertex form, the vertex is visible as (h, k).

What is the axis of symmetry?

The axis of symmetry is the vertical line that passes through the vertex and splits the parabola into two mirror-image halves. For f(x)=ax²+bx+c, the axis is x = -b/(2a). In vertex form, it is x = h.

Why does a positive a mean minimum?

When a is positive, the parabola opens upward like a U-shape. The vertex is the lowest point, so it gives a minimum value. When a is negative, the parabola opens downward, so the vertex is the highest point and gives a maximum value.

Vertex Form Calculator – Convert Standard Form to Vertex Form

Why vertex form is useful

Vertex form shows the turning point of a parabola immediately. Standard form, ax² + bx + c, is often easier for arithmetic and intercepts, but vertex form, a(x - h)² + k, makes the vertex (h, k) visible. OpenStax demonstrates rewriting quadratic functions in vertex form by completing the square in its quadratic transformations lesson.

This calculator uses the equivalent vertex formula h = -b/(2a), then evaluates k = f(h). That gives the same vertex you would get by completing the square, but it is faster when you only need the result.

Worked example

f(x) = x² + 6x + 5\n\nh = -b/(2a) = -6/(2×1) = -3\nk = f(-3) = (-3)² + 6(-3) + 5 = -4\n\nVertex form: f(x) = (x + 3)² - 4\nVertex: (-3, -4)

Because a is positive, the parabola opens upward and the vertex gives the minimum value.

Standard form vs vertex form

Standard form is best for seeing the y-intercept and using the quadratic formula. Vertex form is best for graphing, finding minimum or maximum values, and understanding transformations. A strong quadratic page should connect both forms because students often move back and forth between them.

Common questions

Vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. The value of a controls whether the parabola opens up or down and how narrow or wide it is. Vertex form is especially useful for graphing and optimization problems.
For f(x)=ax²+bx+c, calculate h = -b/(2a), then calculate k = f(h). After that, write the function as f(x)=a(x-h)²+k. You can also convert by completing the square, which shows the algebraic structure behind the formula.
The vertex is the turning point of the parabola. If the parabola opens upward, the vertex is the minimum point. If the parabola opens downward, the vertex is the maximum point. In vertex form, the vertex is visible as (h, k).
The axis of symmetry is the vertical line that passes through the vertex and splits the parabola into two mirror-image halves. For f(x)=ax²+bx+c, the axis is x = -b/(2a). In vertex form, it is x = h.
When a is positive, the parabola opens upward like a U-shape. The vertex is the lowest point, so it gives a minimum value. When a is negative, the parabola opens downward, so the vertex is the highest point and gives a maximum value.