Quadratic equation through three points
Finding a quadratic equation through three points is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students set up and solve a system of equations to determine the coefficients a, b, and c in ax^2 + bx + c that fits three given coordinate pairs.
Key Concepts
Property A quadratic equation can be written in the form $$y = ax^2 + bx + c$$ To find the three parameters $a$, $b$, and $c$, you need three data points. Substituting the coordinates of each of the three points into the equation of the parabola creates a system of three linear equations in the three unknowns $a$, $b$, and $c$, which can then be solved.
Examples To find the parabola through $(1, 3)$, $(3, 5)$, and $(4, 9)$, we solve the system: $a + b + c = 3$, $9a + 3b + c = 5$, and $16a + 4b + c = 9$. The solution is $a=1, b= 3, c=5$, so the equation is $y = x^2 3x + 5$.
A parabola passes through $(0, 8)$, $(1, 5)$, and $(2, 6)$. The system is $c = 8$, $a+b+c=5$, and $4a+2b+c=6$. Substituting $c=8$ gives $a+b= 3$ and $4a+2b= 2$. The solution is $a=2, b= 5, c=8$, so $y = 2x^2 5x + 8$.
Common Questions
How do you find a quadratic equation from three points?
Substitute each point into y = ax^2 + bx + c to create three equations, then solve the resulting 3x3 system for a, b, and c.
Why do you need exactly three points to define a parabola?
A quadratic has three unknown coefficients (a, b, c), so exactly three points give three equations — just enough to find a unique solution.
What if the three points are collinear?
If all three points lie on a straight line, the resulting system will give a = 0, meaning it is actually a linear function, not a quadratic.
Can you use a calculator to find a quadratic through three points?
Yes, most graphing calculators have a quadratic regression or system solver feature that can find the equation given three data points.