- #1
starchild75
- 100
- 1
Homework Statement
Find an equation of a parabola given three points without a vertex point.
Homework Equations
y=a(x-h)+k
The Attempt at a Solution
The parabola is upside down to I know a is negative.
The general form will give you your method. If you have three points, then you can form three equations in general form. The unknowns will be the coefficients. You can simply solve the system of linear equations. This would be introductory level algebra.starchild75 said:I don't know how to find the variables given the three points.
You said you had 3 equations- you've only written 2 there!starchild75 said:the 3 points are: (-2,2), (0,1), (1,-2.5)
I substituted the numbers, giving three equations.
2=4a-2b+c
1=0x^2+0b+C
-5/2=a+b+c
so c=1
-2b=-4a+1
b=(4a+1)/2
I can't get the numbers to work.
This is actually in my calculus book. chapter 1
Sorry, but it's too early to feel lucky! That's not at all right. Putting a= 2, b= 11/2, c= 1 into the first equation, you get 4(2)- 2(11/2)+ 1= 2- 11+ 1= -8, not 2.starchild75 said:I got the numbers to work. The equation in general form is 2x^2-11/2x+1. How would I convert that to standard form. What is the trick to solving these? I feel like I got lucky to get the answer.
starchild75 said:I think I switched the signs around. Now I get
2=4a-2b+c
1=0+0+c
-5/2=a+b+c
and got
a=-1
b=-5/2
c=1
Does that sound better?
Finding the equation of a parabola given three points involves following these steps:
The general equations for a parabola are as follows:
The type of parabola (whether it opens upward or downward) can be determined from the sign of the coefficient "a" in the general equation. If "a" is positive, the parabola opens upward; if "a" is negative, it opens downward.
Yes, you can find the equation of a parabola given three non-collinear (not in a straight line) points. Three non-collinear points uniquely determine a parabola. However, if the three points are collinear (lie on a straight line), they do not uniquely define a parabola, as an infinite number of parabolas can pass through them.
If the three given points are collinear and lie on a straight line, you cannot uniquely determine a parabola that passes through them. In such cases, you may not be able to find a unique equation for the parabola.
One special case to consider is when the three points are all located on the axis of symmetry (vertical line) of the parabola. In this case, the vertex form of the equation can be simplified, making it easier to find the equation. Additionally, if one of the given points is the vertex of the parabola, it simplifies the equation and makes solving for the coefficients a, b, and c more straightforward.