Problem 1 Writing an Equation of a Parabola

A parabola contains the points (0, 0), ( − 1 , − 2 ) , open negative 1 comma negative 2 close comma  and (1, 6). What is the equation of this parabola in standard form?

Substitute the (x, y) values into y = a x 2 + b x + c y equals eh , x squared , plus b x plus c  to write a system of equations.

y = a x 2 + b x + c 0 = a ( 0 ) 2 + b ( 0 ) + c → c = 0 Use  ( 0 , 0 ) . − 2 = a ( − 1 ) 2 + b ( − 1 ) + c → − 2 = a − b + c Use  ( − 2 , − 2 ) . 6 = a ( 1 ) 2 + b ( 1 ) + c → 6 = a + b + c Use  ( 1 , 6 ) . table with 4 rows and 2 columns , row1 column 1 , y , column 2 equals eh , x squared , plus b x plus c , row2 column 1 , 0 , column 2 equals eh open 0 , close squared , plus b open 0 close plus c , column 3 rightwards arrow , column 4 c , column 5 equals 0 , column 6 cap use , open 0 comma 0 close . , row3 column 1 , negative 2 , column 2 equals eh open negative 1 , close squared , plus b open negative 1 close plus c , column 3 rightwards arrow , column 4 negative 2 , column 5 equals eh minus b plus c , column 6 cap use , open negative 2 comma negative 2 close . , row4 column 1 , 6 , column 2 equals eh open 1 , close squared , plus b open 1 close plus c , column 3 rightwards arrow , column 4 6 , column 5 equals eh plus b plus c , column 6 cap use , open 1 comma 6 close . , end table

Since c = 0, the resulting system has two variables.

{ a − b = − 2 ◄ a + b = 6 Use elimination . a = 2  and  b = 4. left brace . table with 2 rows and 1 column , row1 column 1 , eh minus b equals negative 2 black left pointing pointer , row2 column 1 , eh plus b equals 6 , end table . begin box , table with 2 rows and 1 column , row1 column 1 , cap useelimination . . , row2 column 1 , eh equals 2 , and , b equals 4. , end table , end box

Substitute a = 2, b = 4, and c = 0 into standard form: y = 2 x 2 + 4 x + 0 . y equals 2 , x squared , plus 4 x plus 0 .

y = 2 x 2 + 4 x y equals , 2 x squared , plus 4 x  is the equation of the parabola that contains the given points.

✓ Got It?

1. What is the equation of a parabola containing the points (0, 0), ( 1 , − 2 ) , open 1 comma negative 2 close comma  and ( − 1 , − 4 ) ? open negative 1 comma negative 4 close question mark