Vertex Form Calculator

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!

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

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.

If you want to convert a quadratic equation from the standard form to the vertex form, you can use completing the square method (you can read more about it in our completing the square calculator). Let's discuss how this method works in our current context.

To convert the standard form y = ax² + bx + c to vertex form:

1. Extract a from the first two terms: y = a[x² + (b/a)x] + c.

2. Add and subtract (b/(2a))² inside the bracket: y = a[x² + (b/a)x + (b/(2a))² - (b/(2a))²] + c.

3. Use the short multiplication formula: y = a[(x + b/(2a))² - (b/(2a))²] + c.

4. Expand the bracket: y = a(x + b/(2a))² - b²/(4a) + c.

5. This is your vertex form with h = -b/(2a) and k = c - b²/(4a).

That is one way 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.

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, we will give the recipe in the next section.

To convert a parabola from vertex to standard form:

1. Write down the parabola equation in the vertex form:

   y = a(x-h)² + k.

2. Expand the expression in the bracket:

   y = a(x² - 2hx + h²) + k.

3. Multiply the terms in the parenthesis by a:

   y = ax² - 2ahx + ah² + k.

4. Compare the outcome with the standard form of a parabola:

   y = ax² + bx + c.

5. You have the standard from! Its parameters are b = -2·a·h, c = a·h² + k.

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 an accuracy of four decimal places.

If you know the parameters a, b, and c from the standard form of a parabola, you can find the vertex coordinates h and k by using the formulas:

• h = -b/(2a); and
• k = c - b²/(4a).

Alternatively, you can evaluate the value of your parabola at the argument h, i.e., k = ah² + bh + c.

The vertex form is y = a(x - 2)² + 5, where a is the same non-zero parameter as in the standard form. For each value of a, you get a different parabola, so you need to specify a to get a definite result.