How do I find the vertex of a parabola?

Question

How do I find the vertex of the parabola defined by: y=x^2-8x+18? Then how do I find the equation of the line through the vertex and the point, (8,1)??

Thanks

Answer 1

You can always graph it and then see where it is.

Analytically, you need to find the point where the change in slope is 0. This is finding where y' (derivative) = 0

y = x^2 -8x+18 then y' = 2x-8 so x = 4

Now x=4 implies that y = 4^2 -8*4 + 18 = 16-32+18 = 2

Now you need to find a line through two points:
(4,2) and (8,1)

Find the slope and the intercept:

y = mx+ b

m = (y1-y2)/(x1-x2) = (2-1)/(4-8) = -1/4

Substitute back in, and use one of the points to solve for b.

2 = -1/4* 4 + b

implies b = 2+1 = 3

Therefore y = (-1/4)*x + 3

Answer 2

Find the first derivative of the function and set it equal to zero.
Because the slope =0 at the vertex.
Here: y' = 2x -8 =0; solving for x we get x=4
Subst x=4 back into the eq for y we get y= 16-32 +18 = 2
So the vertex point is V(4,2)
Now just find the equation of a line thru these two points (4,2) and (8,1).
Slope of the line =( 2-1)/(4-8)= 1/-2= -1/2
y-1=(-1/2)x+4
y= (-1/2)x +3

Answer 3

The minimum point is your vertex

x^2 - 8x + 18

2x - 8 = 0

x = 4

When x = 4

y = 16 - 32 + 18 = 2

Vertex = (4,2)

Find slope

(2-1) / (4-8) = -1/4

y = -1/4x + b

Plug a point in

1 = -2 + b

b = 3

y = -1/4x + 3

___________________________________________

I used a slight bit of calculus here, so I'll do it the algebraic way.

To find the vertex, the formula is (-b/2a,-b^2/4a+c)

The usual equation is

y= ax^2 + bx + c

By plugging them in you will get your vertex.

I'll show you how to do it.

8 / 2 = 4

x-coordinate = 4

-64 / 4 + 18 = -16 + 18 = 2

y-coordinate = 2

The y-coordinate formula is derived from plugging x into the function.

Answer 4

There are several ways you can do this:

One, you can put the equation into the vertex form.
y=a(x-h)^2 + k
then
(h,k) is the vertex.

Or - you can work with the standard form.
y=ax^2 + bx + c
The axis of symetry is x=-b/2a
Now you know where x will fall.
Plug this value into the equation and figure out the y.

Once this is done, you have two sets of coordinates.
Use the slope formula m=(Y2-Y1)/(X2-X1)
Then use the point formula y-y1=m(x-x1)
then manipulate it into y= format if you need to.

Folks, age/level appropriate answers please. Obviously, this person does not know calculus. Involving derivative and the concept of slope=0 or minima will not help.

Answer 5

NOUN:
pl. ver·ti·ces (-t-sz) KEY also ver·tex·es

1. The highest point; the apex or summit: the vertex of a mountain.
2. Anatomy
1. The highest point of the skull.
2. The top of the head.
3. Astronomy The highest point reached in the apparent motion of a celestial body.
4. Mathematics
1. The point at which the sides of an angle intersect.
2. The point on a triangle or pyramid opposite to and farthest away from its base.
3. A point on a polyhedron common to three or more sides.