Write the general equation of a parabola with its axis parallel to the x-axis.

To find the equation of the parabola with its axis parallel to the x-axis and passing through the given points, we use the general form of such a parabola. Step 1: Write the general equation of a parabola with its axis parallel to the x-axis. The general equation for a parabola with its axis parallel to the x-axis is given by: x = ay^2 + by + c where a, b, and c are constants to be determined. Step 2: Substitute the first point (-1,2) into the equation. Substituting x = -1 and y = 2: -1 = a(2)^2 + b(2) + c -1 = 4a + 2b + c (*1) Step 3: Substitute the second point (1,-1) into the equation. Substituting x = 1 and y = -1: 1 = a(-1)^2 + b(-1) + c 1 = a - b + c (*2) Step 4: Substitute the third point (2,1) into the equation. Substituting x = 2 and y = 1: 2 = a(1)^2 + b(1) + c 2 = a + b + c (*3) Step 5: Solve the system of linear equations for a, b, and c. Subtract equation (2) from equation (3): (a + b + c) - (a - b + c) = 2 - 1 2b = 1 b = (1)/(2) Substitute b = (1)/(2) into equation (*3): a + (1)/(2) + c = 2 a + c = 2 - (1)/(2) a + c = (3)/(2) (*4) Substitute b = (1)/(2) into equation (*1): 4a + 2((1)/(2)) + c = -1 4a + 1 + c = -1 4a + c = -2 (*5) Subtract equation (4) from equation (5): (4a + c) - (a + c) = -2 - (3)/(2) 3a = -(4)/(2) - (3)/(2) 3a = -(7)/(2) a = -(7)/(6) Substitute a = -(7)/(6) into equation (*4): -(7)/(6) + c = (3)/(2) c = (3)/(2) + (7)/(6) c = (9)/(6) + (7)/(6) c = (16)/(6) c = (8)/(3) Step 6: Write the final equation of the parabola. Substitute the values of a = -(7)/(6), b = (1)/(2), and c = (8)/(3) into the general equation x = ay^2 + by + c: x = -(7)/(6)y^2 + (1)/(2)y + (8)/(3) To eliminate fractions, multiply the entire equation by 6: 6x = -7y^2 + 3y + 16 Rearranging the terms to one side: 7y^2 - 3y + 6x - 16 = 0 The equation of the parabola is 7y^2 - 3y + 6x - 16 = 0. Send me the next one 📸