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.
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Here are the solutions for Question 2 and Question 3. Question 2 2.1 a) To find the rule, we observe the relationship between x and y from the given pairs: (-1, -3), (0, -1), (1, 1). The difference in y for a unit increase in x is: -1 - (-3) = 2 1 - (-1) = 2 This indicates a slope of 2. So, the rule is in the form y = 2x + c. Using the point (0, -1): -1 = 2(0) + c -1 = c Thus, the rule is y = 2x - 1. y = 2x - 1 b) The rule states that to get the output value (y), you multiply the input value (x) by 2 and then subtract 1. c) Using the rule y = 2x - 1: To find m: When y = 3, x = m. 3 = 2m - 1 3 + 1 = 2m 4 = 2m m = 2 To find n: When x = 3, y = n. n = 2(3) - 1 n = 6 - 1 n = 5 m = 2, n = 5 d) The ordered pairs are: (-1; -3), (0; -1), (1; 1), (2; 3), (3; 5) e) The relationship as a flow diagram: x Multiply by 2 Result Subtract 1 y 2.2 To determine the rule for the table: | x | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | y | 4 | 6 | 8 | 10 | 12 | The difference in y for a unit increase in x is: 6 - 4 = 2 8 - 6 = 2 This indicates a slope of 2. So, the rule is in the form y = 2x + c. Using the point (1, 4): 4 = 2(1) + c 4 = 2 + c c = 2 Thus, the rule is y = 2x + 2. y = 2x + 2 Question 3 3.1 Complete the table for each equation: For y = -4x: • If x = -1, y = -4(-1) = 4 • If x = 0, y = -4(0) = 0 • If x = 1, y = -4(1) = -4 For y = -4x + 1: • If x = -1, y = -4(-1) + 1 = 4 + 1 = 5 • If x = 0, y = -4(0) + 1 = 0 + 1 = 1 • If x = 1, y = -4(1) + 1 = -4 + 1 = -3 The completed table: | x | -2 | -1 | 0 | 1 | |---|----|----|---|---| | y = -4x | 8 | 4 | 0 | -4 | | y = -4x + 1 | 9 | 5 | 1 | -3 | | Ordered pairs For y = -4x + 1 | (-2; 9) | (-1; 5) | (0; 1) | (1; -3) | 3 done, 2 left today. You're making progress.