Vertex form


Vertex form

Enter the function to be put into the vertex form.

Hint: Enter as 3*x^2.






What is the vertex form?

The vertex form is a special form of a quadratic function. From the vertex form, it is easily visible where the maximum or minimum point (the vertex) of the parabola is: The number in brackets gives (trouble spot: up to the sign!) the x-coordinate of the vertex, the number at the end of the form gives the y-coordinate. This means: If the vertex form is a(x-h)^2+k, then the vertex is at (h|k) .

How to put a function into vertex form?

You have to complete the square: Take the number in front of x, divide it by 2 and square the result. Here is an example:


Mathepower works with this function:

So, the vertex form of your function is f(x)=1*(x+2)^2+-3
The vertex is at (-2|-3)

This is the graph of your function.
Dein Browser unterstützt den HTML-Canvas-Tag nicht. Hol dir einen neuen. :P
  • vertex point at (-2|-3)
This is what Mathepower calculated:
f(x)=1*x^2+4*x+1
f(x)=x^2+4*x+(2)^2+-1*(2)^2+1( Complete the square )
f(x)=(x+2)^2+-1*(2)^2+1( Use the binomial formula )
f(x)=(x+2)^2+-3( simplify )
f(x)=1*(x+2)^2+-3( expand )

As you can see, the x-coordinate of the vertex equals the number in brackets, but only up to change of signs. Furthermore, one sees from this calculation that you just have to use the binomial formula backwards: Build a binomial formula out of the function term. This does only work if there is the right number (the number completing the square). So simply add the right number and subtract it at the same time.

And if there is a number in front of the x^2 ?

Then you have to factor this number out. Example:


Mathepower works with this function:
=3*x^2+-1*24*x+15

So, the vertex form of your function is f(x)=3*(x+-4)^2+-33
The vertex is at (4|-33)

This is the graph of your function.
Dein Browser unterstützt den HTML-Canvas-Tag nicht. Hol dir einen neuen. :P
  • vertex point at (4|-33)
This is what Mathepower calculated:
f(x)=3*x^2+-24*x+15
f(x)=3*(x^2+-8*x+5)( Factor out )
f(x)=3*(x^2+-8*x+(-4)^2+-1*(-4)^2+5)( Complete the square )
f(x)=3*((x+-4)^2+-1*(-4)^2+5)( Use the binomial formula )
f(x)=3*((x+-4)^2+-11)( simplify )
f(x)=3*(x+-4)^2+-33( expand )

It is important to factor out first and complete the square afterwards. Otherwise there could be nasty mistakes. (Unfortunately, many people do not think about such stuff and simply use the binomial formula even if it is not possible… More unfortunately, terms cannot cry ""OUCH!"", but just math teachers can when they see such a calculation.)

And if there is a minus in front of the x^2 ?

Simply factor -1 out. Btw: Whenever there is a negative number in front of the x^2, the parabola is open downward. Example:


Mathepower works with this function:
=-1*x^2+-1*3*x+2

So, the vertex form of your function is f(x)=-1*(x+3/2)^2+17/4
The vertex is at (-3/2|17/4)

This is the graph of your function.
Dein Browser unterstützt den HTML-Canvas-Tag nicht. Hol dir einen neuen. :P
  • vertex point at (-1.5|4.25)
This is what Mathepower calculated:
f(x)=-1*x^2+-3*x+2
f(x)=-1*(x^2+3*x+-2)( Factor out )
f(x)=-1*(x^2+3*x+(3/2)^2+-1*(3/2)^2+-2)( Complete the square )
f(x)=-1*((x+3/2)^2+-1*(3/2)^2+-2)( Use the binomial formula )
f(x)=-1*((x+3/2)^2+1*-17/4)( simplify )
f(x)=-1*(x+3/2)^2+17/4( expand )

And how is the general formula for the vertex point?

No problem for Mathepower. Simply enter the function f_f(x)=ax^2+bx+c .


Mathepower works with this function:

So, the vertex form of your function is f(x)=1*a*(x+b/(2*a))^2+(1*a*c+-0.25*b^2)/a
The vertex is at ((-1*b)/(2*a)|(1*a*c+-0.25*b^2)/a)

This is what Mathepower calculated:
f(x)=1*a*x^2+1*b*x+1*c
f(x)=1*a*(x^2+b/a*x+c/a)( Factor out )
f(x)=1*a*(x^2+b/a*x+(b/(2*a))^2+-1*(b/(2*a))^2+c/a)( Complete the square )
f(x)=1*a*((x+b/(2*a))^2+-1*(b/(2*a))^2+c/a)( Use the binomial formula )
f(x)=1*a*((x+b/(2*a))^2+1*(1*a*c+-0.25*b^2)/a^2)( simplify )
f(x)=1*a*(x+b/(2*a))^2+(1*a*c+-0.25*b^2)/a( expand )

Can I see even more examples?

Of course. This is a free vertex form calculator. Just enter your example and it will be solved.