transform the ff. into the form f(x) = a(x - h)2 + k:
f(x)=x2-6x+4
f(x)=x2+4x+2
f(x)=3x2-5x+1
tnx a bunch!!!1
I'll do the third one for you, the first two are really easy
f(x)=3x^2-5x+1
=3(x^2 - 5/3 x ) + 1 factored out the 3 from first 2 terms
=3(x^2 - 5/3 x + 25/36 - 25/36) + 1 took half the coefficient of the x term, squared it, then added and subtracted it
= 3((x-5/6)^2 - 25/36) + 1 changed x^2 - 5/3 x + 25/36 to (x-5/6)^2, that is why it is called "completing the square"
=3(x-5/6)^2 - 25/12 + 1 multiplied by 3
=3(x-5/6)^2 - 13/12
of course a quicker way would be to find the x of the vertex using x=-b/(2a) = 5/(2*3) = 5/6 then subbingh that back into the equation to get y.
y = 3(25/36) - 5(5/6)+1
=25/12 - 25/6 + 1
=(25 - 50 + 12)/12
= -13/12
now form your equation
y = 3(x-5/6)^2 - 13/12
To transform the given quadratic functions into the form f(x) = a(x - h)^2 + k, where a, h, and k are constants:
1. f(x) = x^2 - 6x + 4
To complete the square, follow these steps:
a) Take half of the coefficient of the x term and square it: (-6/2)^2 = 9.
b) Add and subtract the value obtained in step a inside the parentheses: x^2 - 6x + 9 - 9 + 4.
c) Group the first three terms and rewrite them as a perfect square trinomial: (x^2 - 6x + 9) - 9 + 4.
d) Simplify the expression: (x - 3)^2 - 5.
e) The equation is now in the desired form: f(x) = (x - 3)^2 - 5.
2. f(x) = x^2 + 4x + 2
Follow the same steps as above:
a) Take half of the coefficient of the x term and square it: (4/2)^2 = 4.
b) Add and subtract the value obtained in step a inside the parentheses: x^2 + 4x + 4 - 4 + 2.
c) Group the first three terms and rewrite them as a perfect square trinomial: (x^2 + 4x + 4) - 4 + 2.
d) Simplify the expression: (x + 2)^2 - 2.
e) The equation is now in the desired form: f(x) = (x + 2)^2 - 2.
3. f(x) = 3x^2 - 5x + 1
Following the same process as above:
a) Take half of the coefficient of the x term and square it: (-5/2)^2 = 6.25.
b) Add and subtract the value obtained in step a inside the parentheses: 3x^2 - 5x + 6.25 - 6.25 + 1.
c) Group the first three terms and rewrite them as a perfect square trinomial: (3x^2 - 5x + 6.25) - 6.25 + 1.
d) Simplify the expression: 3(x - 5/6)^2 - 13/12.
e) The equation is now in the desired form: f(x) = 3(x - 5/6)^2 - 13/12.
Thus, the given quadratic functions have been transformed into the desired form.