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Angelina_Jolie [31]
3 years ago
11

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

Mathematics
1 answer:
Marta_Voda [28]3 years ago
3 0

Answer:

  • vertex form: f(x) = a(x -h)^2 +k
  • standard form: f(x) = ax^2 +bx +c

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 <u>vertex form</u>, 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 ...

  • a = a
  • b = -2ah
  • c = ah^2 +k

__

In <u>standard form</u>, 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)

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(x+y)ⁿ = ∑ ⁿCₐxⁿ⁻ᵃyᵃ

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