use quadratic regression to find the equation for the parabola going through these 3 points. (2, 19), (6…

Explanation:

A quadratic equation has the form ( y = ax^2 + bx + c ). We can substitute each of the three points into this equation to create a system of equations and then solve for ( a ), ( b ), and ( c ).

Step 1: Substitute the point ((2, 19))

Substitute ( x = 2 ) and ( y = 19 ) into ( y = ax^2 + bx + c ): [ 19 = a(2)^2 + b(2) + c ] [ 19 = 4a + 2b + c \quad (1) ]

Step 2: Substitute the point ((6, 99))

Substitute ( x = 6 ) and ( y = 99 ) into ( y = ax^2 + bx + c ): [ 99 = a(6)^2 + b(6) + c ] [ 99 = 36a + 6b + c \quad (2) ]

Step 3: Substitute the point ((-1, 1))

Substitute ( x = -1 ) and ( y = 1 ) into ( y = ax^2 + bx + c ): [ 1 = a(-1)^2 + b(-1) + c ] [ 1 = a - b + c \quad (3) ]

Step 4: Subtract equation (1) from equation (2)

[ (36a + 6b + c) - (4a + 2b + c) = 99 - 19 ] [ 32a + 4b = 80 ] Divide both sides by 4: [ 8a + b = 20 \quad (4) ]

Step 5: Subtract equation (3) from equation (1)

[ (4a + 2b + c) - (a - b + c) = 19 - 1 ] [ 3a + 3b = 18 ] Divide both sides by 3: [ a + b = 6 \quad (5) ]

Step 6: Subtract equation (5) from equation (4)

[ (8a + b) - (a + b) = 20 - 6 ] [ 7a = 14 ] Divide both sides by 7: [ a = 2 ]

Step 7: Substitute ( a = 2 ) into equation (5)

[ 2 + b = 6 ] Subtract 2 from both sides: [ b = 4 ]

Step 8: Substitute ( a = 2 ) and ( b = 4 ) into equation (3)

[ 1 = 2 - 4 + c ] [ 1 = -2 + c ] Add 2 to both sides: [ c = 3 ]

Answer:

The quadratic equation is ( y = 2x^2 + 4x + 3 ), so the coefficients are ( 2 ) (for ( x^2 )), ( 4 ) (for ( x )), and ( 3 ) (constant term). Thus, the filled equation is ( y = \boxed{2}x^2 + \boxed{4}x + \boxed{3} ).