882 CHAPTER 9 Systems and Matrices EXAMPLE 8 Using Modeling to Find an Equation through Three Points Find an equation of the parabola y = ax2 + bx + c that passes through the points 12, 42, 1-1, 12, and 1-2, 52. SOLUTION The three ordered pairs represent points that lie on the graph of the given equation y = ax2 + bx + c, so they must all satisfy the equation. Substituting each ordered pair into the equation gives three equations with three unknowns. 4 = a1222 + b122 + c, or 4 = 4a + 2b + c (1) 1 = a1-122 + b1-12 + c, or 1 = a - b + c (2) 5 = a1-222 + b1-22 + c, or 5 = 4a - 2b + c (3) To solve this system, first eliminate c using equations (1) and (2). 4 = 4a + 2b + c (1) -1 = -a + b - c Multiply (2) by -1. 3 = 3a + 3b (4) Now, use equations (2) and (3) to eliminate the same unknown, c. 1 = a - b + c (2) -5 = -4a + 2b - c Multiply (3) by -1. -4 = -3a + b (5) Solve the system of equations (4) and (5) in two unknowns by eliminating a. 3 = 3a + 3b (4) -4 = -3a + b (5) -1 = 4b Add. - 1 4 = b Divide by 4. Find a by substituting - 1 4 for b in equation (4). 1 = a + b Equation (4) divided by 3 1 = a + a- 1 4b Let b = - 1 4 . 5 4 = a Add 1 4 . Finally, find c by substituting a = 5 4 and b = - 1 4 in equation (2). 1 = a - b + c (2) 1 = 5 4 - a- 1 4b + c Let a = 5 4 , b = - 1 4 . 1 = 6 4 + c Add. - 1 2 = c Subtract 6 4 . The required equation is y = 5 4 x 2 - 1 4 x - 1 2, or y = 1.25x 2 - 0.25x - 0.5. S Now Try Exercise 79. Equation (5) must have the same two unknowns as equation (4). −10 −10 10 10 This graph/table screen shows that the points 12, 42, 1-1, 12, and 1-2, 52 lie on the graph of y1 = 1.25x2 - 0.25x - 0.5. This supports the result of Example 8.
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