How do you write a quadratic function in vertex and standard form

Step-by-step explanation:

Often, you're given one form and asked to write the equation in the other form. Here, we can show the relationship between the two forms.

In vertex form, the vertex of the function (h, k) is obvious in the way the function expression is written:

  f(x) = a(x -h)^2 +k . . . . . . . for vertex (h, k) and vertical scale factor "a"

If we "simplify" this form, we get ...

  f(x) = a(x^2 -2hx +h^2) +k

  f(x) = ax^2 -2ah + (ah^2 +k) . . . . . "standard form" from vertex form

Comparing this to standard form, we can see the relations between the coefficients are ...

__

In standard form, terms are written in descending order of the exponent of the variable.

  f(x) = ax^2 +bx +c

Generally, coefficients are named in alphabetical order, starting with "a" for the leading coefficient (the coefficient of the highest-degree term).

We can use the relations shown above to find the vertex from from these coefficients.

  b = -2ah

  h = -b/(2a) . . . . . divide by the coefficient of h

And the other coefficient of the vertex is ...

  k = c - ah^2 . . . . subtract ah^2 from the equation for c

  k = c - b^2/(4a)

Then ...

  f(x) = a(x +b/(2a))^2 +(c -b^2/(4a)) . . . . . "vertex form" from standard form

You may notice that the key relationship is that between "b" and "h". It is useful to remember it:

  h = -b/(2a)