Question 1829: How do you find the equation of the parabola passing through the points (3,1), (2,3), and (0,-5) Found 2 solutions by longjonsilver, Ne0:Answer by longjonsilver(2297) (Show Source):
You can put this solution on YOUR website! You have to substitute in the x,y values you know into the general equation of a quadratic (parabola), which is and then solve the 3 equation to find the 3 unknowns, a,b,c. Tedious but not difficult...
the third equation helps enormously...we know c!. Put this into the first 2 equations and find a and b.
You can do that :-). Don't forget to write out the equation, once you know the values of a, b and c!.
cheers
jon
You can put this solution on YOUR website! Well you want to start off first by restating the general form of a parabola. . We will be using this as a guide to model the parabola that we want to create.
We will start off with the pair of coordinant points (0,-5). By substituting the points into the x and y of the equation we get . By eliminating all the zero terms we get c=-5. Now you can see why we started off with the coordinant points (0,-5).
We can now use any coordinant points from here on out. I'll choose the coordinant points (2,3). You follow the same steps that you previously did with the substitution. Remember your value of c=-5. Well don't forgot that as we'll be substituting that into all values of c from here on out. We get . Simplifying this and adding 5 to both sides of the equation we get .
Now to use the remaining coordinant points (3,1). Substituting the points into the x and y of the equation we get . Simplifying this and adding 5 to both sides we get .
Well what do you do from here on out you might ask. We have two equations both with two unknown coefficients. We can do many things to solve for these such as elimination or substitution. I'll use substitution for this particular time but what you do is up to you. Just to remind you the two equations that we are dealing with right now are and . Lets go with the latter first. We start off by trying to get one of the coeeficients by themselves and then everything else on the other side of the equation. We can subtract a 9a from both sides to get 3b=6-9a. We then divide both sides to get b=2-3a. The next step is to take this and substitute it into the b coefficient of the remaining equation like so: 4a+2(2-3a)=8. We now have an equation with only one unknown coefficient that we can solve for. By distributing we come to 4a+4-6a=8. Combining like terms we get -2a=4 and finally dividing both sides by -2 we arrive with a=-2. Since we have the a coefficient we can back substitute to find our value for b like so: 3b=6-9(-2). By simplifying we get 3b=24 and finally by diving both sides by 3 we get b=8. We have all our coefficients now and are ready to rewrite our equation for the parabola using these terms . We can make sure this is the right parabola by simply graphing it and seeing if it does indeed cross through all of those points.
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