Equation of a Parabola
Date: 12/20/2001 at 17:22:18 From: Richard DeVries Subject: Equation of a parabola Given several points, how do you determine the equation? Assuming that the points appear to be a parabola, how do you approximate the equation that would give the similar graph? Is there a way to be exact?
Date: 12/21/2001 at 05:15:27 From: Doctor Jeremiah Subject: Re: Equation of a parabola Hi Richard, The equation for a parabola is y = ax^2 + bx + c So if you have three points, you can calculate the equation of the parabola. But you need three points. Let's say the points are (-4,15) (2,3) and (1,-5). Notice that these points have an x value and a y value, and the equation has a place for x and a place for y. So for each point you could plug in the the x and y values. (-4,15): y = ax^2 + bx + c <=== x=-4, y=15 15 = a(-4)^2 + b(-4) + c 15 = 16a - 4b + c (2,3): y = ax^2 + bx + c <=== x=2, y=3 3 = a(2)^2 + b(2) + c 3 = 4a + 2b + c (1,-5): y = ax^2 + bx + c <=== x=1, y=-5 -5 = a(1)^2 + b(1) + c -5 = a + b + c Now we have these three equations: 15 = 16a - 4b + c 3 = 4a + 2b + c -5 = a + b + c There are three equations and three unknowns, so if we can solve these equations for a, b, and c, we will be able to turn y = ax^2 + bx + c into the equation for the graph. Step 1: pick an equation at random and solve for one unknown -5 = a + b + c c = -5 - a - b Step 2: plug that into both of the other equations 15 = 16a - 4b + c <=== c = -5 - a - b 15 = 16a - 4b + (-5 - a - b) 15 = 15a - 5b - 5 20 = 15a - 5b 3 = 4a + 2b + c <=== c = -5 - a - b 3 = 4a + 2b + (-5 - a - b) 3 = 3a + b - 5 8 = 3a + b Now we have only two equations with two unknowns: 20 = 15a - 5b 8 = 3a + b Step 3: pick an equation at random and solve for one unknown 8 = 3a + b b = 8 - 3a Step 4: plug that into the other equation 20 = 15a - 5b <=== b = 8 - 3a 20 = 15a - 5(8 - 3a) 20 = 15a - 40 + 15a 60 = 15a + 15a 60 = 30a 60/30 = a a = 2 Step 4: plug that into the answer for step 3 b = 8 - 3a <=== a=2 b = 8 - 3(2) b = 2 Step 5: plug both of those into the answer from Step 1 c = -5 - a - b <=== a=2, b=2 c = -5 - (2) - (2) c = -9 The equation for a parabola is y = ax^2 + bx + c and we have values for a, b, and c, so: y = ax^2 + bx + c <=== a=2, b=2, c=-9 y = 2x^2 + 2x - 9 And that's the equation of our parabola. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/
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