Parabola question?

Question

I have no clue where to start here.

if we know that (3.5,-30.25) is the vertex of the parabola below, figure out 3 more points that would also be on the parabola. Explain how you know that your 3 points are on this parabola even though we don't know it's quadratic equation.

Other points already given on parabola besides the (3.5,-30.25) are (4,-30) (6,-24) (10,12)

I'm stressing out trying to solve this.

Answer 1

We have a parabola in the form

ax² + bx + c = y

Plug in the three points (4,-30); (6,-24); and (10,12) and get three equations in three unknowns.

16a + 4b + c = -30
36a + 6b + c = -24
100a + 10b + c = 12

Solving we get:

a = 1
b = -7
c = -18

The equation of the parabola is:

y = x² - 7x - 18

As a check, let's see if we get the vertex. Complete the square.

y = [x² - 7x + (7/2)²] - 18 - (7/2)²
y = (x - 7/2)² - (18 + 49/4)
y = (x - 3.5)² - 30.25

This is the vertex we expected.

Answer 2

The vertex is not on the y-axis but the parabola opens up towards the positive y-axis. So the equation of the parabola is
(y-a) = (x-b)^2 + c where a, b are constants and c is the y-intercept
We have four equations and four unknowns:
-30.25 - a = (3.5-b)^2 + c
-30 - a = (4 -b)^2 + c
-24 -a = (6-b)^2 + c
12 - a = (10-b)^2 + c
because you may have to solve for b and b^2 separately

Now find a, b, c, plug into the general eqn and you can find any point on y(x)

Answer 3

parabolas have a line of symmetry that runs through the vertex

so you can figure out 3 symmetrical points from the other points given

you are given ( 4, -30 )

since 4 is .5 to the right of 3.5

the symmetrical value to the left is 3 ( .5 to the left of 3.5 )

in order for it to be symmetrical the y part of the pair for ( 3 , ? ) would have to be the same as ( 4, - 30 )

so the pair would have to be ( 3, -30 )

if it wasn't - 30 then the point would be at a different height

you would use the same reasoning to find the other two symmetrical points from the last two given points

Answer 4

first find the equation of parabola -> y2=4ax
than sub the points in this eqn , if they satisfy than they r passing on the parabola