Parabolas

Parabola From Three Points

Fit y = ax^2 + bx + c through three points using exact linear equations.

Example

Fit a quadratic model through three points.

highlighted = computed this step

Step 1 — Given points

Read the three points.

A=(-1,6)B=(0,3)C=(2,3)A=( \hl{-1} , \hl{6} ) \quad B=( \hl{0} , \hl{3} ) \quad C=( \hl{2} , \hl{3} )

Step 2 — Use point A

Substitute point A.

a−b+c=6a-b+c= \hl{6}

Step 3 — Use point B

Substitute point B.

c=3c= \hl{3}

Step 4 — Use point C

Substitute point C.

4a+2b+c=34 a+ 2 b+c= \hl{3}

Step 5 — Solve coefficients

Solve for the coefficients.

a=1,b=-2,c=3a= \hl{1} , b= \hl{-2} , c= \hl{3}

Step 6 — Model

Write the quadratic model.

y=x2−2x+3y= x^{2} - \hl{2} x + \hl{3}

Step 7 — Plot fitted parabola

Plot the fitted parabola.

Coordinate plot\text{Coordinate plot}
Parabola through three points plotParabola through A(-1, 6), B(0, 3), and C(2, 3).ABC

Step 8 — Verify

Check all three points.

all three points satisfy the model\text{all three points satisfy the model}
parabola-from-three-points Substitute each point into y = ax^2 + bx + c, solve for a, b, and c, then verify all three points.