y = a(x - h)² + k Substitute (x,y) = (1,-1) -1 = a(1 - h)² + k Substitute (x,y) = (2,-1) -5 = a(2 - h)² + k Substitute (x,y) = (3,-7) -7 = a(3 - h)² + k Solve this system: -1 = a(1 - h)² + k -5 = a(2 - h)² + k -7 = a(3 - h)² + k Solve the first for k -1 = a(1 - h)² + k -1 - a(1 - h)² = k Substitute in the second and simplify -5 = a(2 - h)² + k -5 = a(2 - h)² - 1 - a(1 - h)² -4 = a(2 - h)² - a(1 - h)² -4 = a[(2-h)² - (1-h)²] -4 = a[(2-h) - (1-h)][(2-h) + (1-h)] -4 = a[2 - h - 1 + h][2 - h + 1 - h] -4 = a[1][3 - 2h] -4 = a(3 - 2h) Substitute in the thirdd and simplify -7 = a(3 - h)² + k -7 = a(3 - h)² - 1 - a(1 - h)² -6 = a(3 - h)² - a(1 - h)² -6 = a[(3-h)² - (1-h)²] -6 = a[(3-h) - (1-h)][(3-h) + (1-h)] -6 = a[3 - h - 1 + h][3 - h + 1 - h] -6 = a[2][4 - 2h] -6 = 2a(4 - 2h) -6 = 2a·2(2 - h) -6 = 4a(2 - h) Divide equals by equals: Cross-multiply: 8(2-h) = 3(3-2h) 16-8h = 9-6h -2h = -7 h = Sustitute in -1 = a(1 - h)² + k -5 = a(2 - h)² + k -1 = a(1 - )² + k -5 = a(2 - )² + k -1 = a + k -1 = a + k Multiply both equations through by 4 -4 = 25a + 4k -20 = 9a + 4k Multiply the 1st eq. by -1 and add term by term 4 = -25a - 4k -20 = 9a + 4k --------------- -16 = -16a 1 = a Substitute in -4 = 25a + 4k -4 = 25(1) + 4k -4 = 25 + 4k -29 = 4k = k So the equation of the parabola in standard form is y = a(x - h)² + k y = 1(x - )² - y = (x - )² - Edwin
