This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Analyze the given data to find a pattern. The given mapping is: x: 1, 2, 3, 4, 5 y: 3, 8, 15, 24, 35 Calculate the first differences in y values: 8 - 3 = 5 15 - 8 = 7 24 - 15 = 9 35 - 24 = 11 The first differences are 5, 7, 9, 11. Step 2: Calculate the second differences. 7 - 5 = 2 9 - 7 = 2 11 - 9 = 2 Since the second differences are constant and equal to 2, the rule is a quadratic function of the form y = ax^2 + bx + c. The second difference is equal to 2a. 2a = 2 a = 1 Step 3: Substitute the value of a into the general quadratic equation and use two points from the table to find b and c. The equation becomes y = x^2 + bx + c. Using the point (x=1, y=3): 3 = (1)^2 + b(1) + c 3 = 1 + b + c b + c = 2 (*) Using the point (x=2, y=8): 8 = (2)^2 + b(2) + c 8 = 4 + 2b + c 2b + c = 4 (**) Step 4: Solve the system of linear equations for b and c. Subtract equation () from equation (*): (2b + c) - (b + c) = 4 - 2 b = 2 Substitute b=2 into equation (*): 2 + c = 2 c = 0 Step 5: Write the final rule of the mapping. Substitute a=1, b=2, and c=0 into y = ax^2 + bx + c: y = 1x^2 + 2x + 0 y = x^2 + 2x The rule of the mapping is y = x^2 + 2x.