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Use vertex form
Vertex form: y = a(x-h)² + k
a
h
k
Results
Vertex form equation:

y = x²

Standard form equation:

y = x²

Characteristic points:

Vertex P(0, 0)

Y-intercept Y(0, 0)
Zero X₁(0, 0)

This is the vertex form calculator (also known as vertex calculator, or even find the vertex calculator). If you want to know how to find the vertex of a parabola, this is the right place to begin. Moreover, our tool teaches you what the vertex form of a quadratic equation is and how to derive the equation of the vertex form or the vertex equation itself.

And this is not the end! This calculator also helps you convert from the standard to the vertex form of a parabola, or even the other way round, in a blink of an eye!

How to find the vertex of a parabola? Vertex equation

The vertex of a parabola is a point that represents the extremal value of a quadratic curve. The quadratic part stands because the most significant power of our variable (x) is two. The vertex can be either a minimum (for a parabola opening up) or a maximum (for a parabola opening down).

Alternatively, we can say that the vertex is the intersection of the parabola and its symmetry axis.

Typically, we denote the vertex as a point P(h,k), where h stands for the x-coordinate, and k indicates the y-coordinate.

That's enough on the definitions. But how to find the vertex of a quadratic function? It may be a surprise, but we don't need to evaluate any square root to do so!

Whenever we face a standard form of a parabola y = a*x² + b*x + c, we can use the equations of the vertex coordinates:

h = -b/(2a),
k = c - b²/(4a).

Knowing these ratios, we can move one step further and ask: What is the vertex form of a parabola?

What is the vertex form of a quadratic equation?

Intuitively, the vertex form of a parabola is the one that includes the vertex’s details inside. We can write the vertex form equation as:

y = a*(x-h)² + k.

As you can see, we need to know three parameters to write a quadratic vertex form. One of them is a, the same as in the standard form. It tells us whether the parabola is opening up (a > 0) or down (a < 0). The parameter a can never equal zero for a vertex form of a parabola (or any other form, strictly speaking).

The remaining parameters, h, and k are the components of the vertex. That's where the vertex form equation gets its name.

Additionally, it's worth mentioning that it's possible to draw a quadratic function graph having only the parameter a and the vertex.

How to convert from the standard form to the vertex form?

We can try to convert a quadratic equation from the standard form to the vertex form using completing the square method:

  1. Write the parabola equation in the standard form: y = a*x² + b*x + c;

  2. Extract a from the first two terms: y = a * (x² + b/a * x) + c;

  3. Complete the square for the expressions with x. The missing fraction is (b/(2a))². Add and subtract this term in the parabola equation: y = a * [x² + b/a * x + (b/(2a))² - (b/(2a))²] + c;

  4. We can compress the three leading terms into a shortcut version of multiplication: y = a * [(x + b/(2a))² - (b/(2a))²] + c;

  5. Remove the square bracket by multiplying the terms by a: y = a*(x + b/(2a))² - b²/(4a) + c;

  6. Compare the outcome with the vertex form of a quadratic equation: y = a*(x-h)² + k;

  7. As a result of the comparison, we know how to find the vertex of a parabola: h = -b/(2a), and k = c - b²/(4a).

That is one way of how to convert to vertex form from a standard one. The second (and quicker) one is to use our vertex form calculator - the way we strongly recommend! It only requires typing the parameters a, b, and c. Then, the result appears immediately at the bottom of the calculator space.

Vertex form to standard form converter

Our find the vertex calculator can also work the other way around by finding the standard form of a parabola. In case you want to know how to do it by hand using the vertex form equation, this is the recipe:

  1. Write the parabola equation in the vertex form: y = a*(x-h)² + k;

  2. Expand the expression in the bracket: y = a*(x² - 2*h*x + h²) + k;

  3. Multiply the terms in the parenthesis by a: y = a*x² - 2*a*h*x + a*h² + k;

  4. Compare the outcome with the standard form of a parabola: y = a*x² + b*x + c;

  5. Estimate the values of parameters: b = -2*a*h, c = a*h² + k.

How to use the vertex form calculator?

There are two approaches you can take to use our vertex form calculator:

  • The first possibility is to use the vertex form of a quadratic equation;

  • The second option finds the solution of switching from the standard form to the vertex form.

We've already described the last one in one of the previous sections. Let's see what happens for the first one:

  • Type the values of parameter a, and the coordinates of the vertex, h and k. Let them be a = 0.25, h = -17, k = -54;

  • That's all! As a result, you can see a graph of your quadratic function, together with the points indicating the vertex, y-intercept, and zeros.

Below the chart, you can find the detailed descriptions:

  • Both the vertex and standard form of the parabola: y = 0.25(x + 17)² - 54 and y = 0.25x² + 8.5x + 18.25 respectively;

  • The vertex: P = (-17, -54);

  • The y-intercept: Y = (0, 18.25);

  • The values of the zeros: X₁ = (-31.6969 , 0), X₂ = (-2.3031, 0). In case you're curious, we round the outcome to five significant figures here.

Wojciech Sas, PhD candidate