how do you use the vertex formula to find the coordinates of the vertex of the graph of a parabola?

Question

how do you use the vertex formula to find the coordinates of the vertex of the graph of a parabola?

Answer 1

How do you use the vertex formula to find the coordinates of the vertex of the graph of a parabola?

 

A simple parabola: y = a x^2 with vertex at (0,0);

 

translated by <h,k>:

 

y - k = a(x - h)^2 with vertex at (h,k), or

 

y = a(x - h)^2 + k with vertex at (h,k).

 

If you have a vertex form equation just pick out the h and k for the vertex coordinates.

 

E.g.,

y = -3(x+7)^2 - 11, or

y = -3(x-(-7))^2 + (-11),

has a vertex at (-7,-11).

 

BTW, where does h = -b/(2a) come from?

 

Multiply out the general vertex form:

 

y = a(x - h)^2 + k

 

y = a(x^2 - 2hx + h^2) + k

 

y = a x^2 + (-2ah)x + (ah^2 + k)

 

Now compare to standard form:

 

y = a x^2 + b x + c

 

and it's easily seen that:

 

b = -2ah => h = -b/(2a), and

 

c = ah^2 + k => k = c - ah^2

 

You can also solve the general vertex form for its zeros to get an alternative quadratic formula:

 

y = a(x - h)^2 + k = 0

 

(x - h)^2 = -k/a

 

|x - h| = √(-k/a) <= imaginary if k and a have same signs

 

x = h ± √(-k/a)

 

If -k/a > 0 then 2 real roots,

if -k/a = 0 then 1 real root, or

if -k/a < 0 then 2 complex conjugate roots.

Answer 2

Hi Ness;

A parabola formula is...

y=ax²+bx+c

vertex is -b/2a

This is easy to understand.

The vertex is the point at which the change of slope is zero.  Let's take the first derivative of the equation and set it equal to zero...

0=2ax+b

-b=2ax

-b/2a=x