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Find the Formula Associated with Each parabola

the question is:

part of a roller coaster is modeled by three parabola - some parts removed to give the diagram below.

The first parabola from the left meets the second at co-ordinates (7, 14). using this information from the diagram how can i find the formula associated with each parabola and also workout their relevant domains? hint: think of the 3rd parabola is a translation of the first.

Attachment 24406

could someone explain in steps if possible how the formulaes and the domains be obtained?

Re: Find the Formula Associated with Each parabola

To find the equation of a parabola passing through three given points, substitute the points' coordinates (x, y) into the equation y = ax^2 + bx + c. For each point (x, y) you'll get an equation on a, b and c. Then solve this system of three equations to find a, b, c.

Re: Find the Formula Associated with Each parabola

vertex form equation of a quadratic ...

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

where $\displaystyle (h,k)$ is the vertex, and the value of the constant $\displaystyle a$ can be determined by substituting any other point $\displaystyle (x,y)$ that lies on the parabola.

Re: Find the Formula Associated with Each parabola

Quote:

Originally Posted by

**emakarov** To find the equation of a parabola passing through three given points, substitute the points' coordinates (x, y) into the equation y = ax^2 + bx + c. For each point (x, y) you'll get an equation on a, b and c. Then solve this system of three equations to find a, b, c.

max(4.5, 20.25)-->y=ax^2+bx+c-->a(4.5)^2+b(4.5)+c

-->20.25a+4.5b+c=14

min(12, 1.5)-->y=ax^2+bx+c-->a(12)^2+b(12)+c

-->144a+12b+c=1.5

max(19.5, 20.25)-->=ax^2+bx+c-->a(19.5)^2+b(19.5)+…

-->380.25a+19.5b+c=20.25

is this correct? what am i meant to do with the co-ordinates (7, 14)? and how do i proceed from this step to obtain not just the formula but ALSO the relevant domains?