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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For the following set of ordered pairs, determine if the mapping is linear or exponential: - (1, 2), (2, 4), (3, 8), (4, 16) Justify your answer. Step 1: Check for a constant difference in the y-values. The y-values are 2, 4, 8, 16. Differences: 4 - 2 = 2 8 - 4 = 4 16 - 8 = 8 Since the differences are not constant (2, 4, 8), the mapping is not linear. Step 2: Check for a constant ratio in the y-values. Ratios: (4)/(2) = 2 (8)/(4) = 2 (16)/(8) = 2 Since there is a constant ratio of 2 between consecutive y-values, the mapping is exponential. The mapping is exponential because there is a constant ratio of 2 between consecutive y-values. Exponential --- 15. Given a linear mapping with the rule y = 3x + 2, calculate the output values for x = 0, 1, 2, and 3. Organize your results in a table. Step 1: Calculate output for x = 0. y = 3(0) + 2 = 0 + 2 = 2 Step 2: Calculate output for x = 1. y = 3(1) + 2 = 3 + 2 = 5 Step 3: Calculate output for x = 2. y = 3(2) + 2 = 6 + 2 = 8 Step 4: Calculate output for x = 3. y = 3(3) + 2 = 9 + 2 = 11 Step 5: Organize results in a table. |c|c| x & y \\ 0 & 2 \\ 1 & 5 \\ 2 & 8 \\ 3 & 11 \\ Table above --- 16. A mobile phone plan charges a fixed fee of GH₵10 plus GH₵0.50 per minute. Write the rule for this mapping in the form y = mx + c, where x represents minutes and y represents total cost. Step 1: Identify the fixed fee (c). The fixed fee is GH₵10, so c = 10. Step 2: Identify the rate per minute (m). The charge per minute is GH₵0.50, so m = 0.50. Step 3: Write the rule in the form y = mx + c. Substituting the values of m and c: y = 0.50x + 10 The rule is y = 0.50x + 10. --- 17. Identify the constant difference in the domain and co-domain for the following mapping, then determine the rule: - Domain: 2, 4, 6, 8 - Co-domain: 7, 13, 19, 25 Step 1: Identify the constant difference in the domain (x-values). Differences: 4 - 2 = 2 6 - 4 = 2 8 - 6 = 2 The constant difference in the domain is 2. Step 2: Identify the constant difference in the co-domain (y-values). Differences: 13 - 7 = 6 19 - 13 = 6 25 - 19 = 6 The constant difference in the co-domain is 6. Step 3: Determine the slope (m). The slope m = change in ychange in x = (6)/(2) = 3. Step 4: Determine the y-intercept (c) using one ordered pair and the slope. Using the point (2, 7) and m = 3 in y = mx + c: 7 = 3(2) + c 7 = 6 + c c = 7 - 6 c = 1 Step 5: Write the rule. The rule is y = 3x + 1. The constant difference in the domain is 2. The constant difference in the co-domain is 6. The rule is y = 3x + 1. --- 18. For the mapping y = 2x + 1, create a table of values for x = -2, -1, 0, 1, 2 and explain what the values of m and c represent. Step 1: Calculate output for x = -2. y = 2(-2) + 1 = -4 + 1 = -3 Step 2: Calculate output for x = -1. y = 2(-1) + 1 = -2 + 1 = -1 Step 3: Calculate output for x = 0. y = 2(0) + 1 = 0 + 1 = 1 Step 4: Calculate output for x = 1. y = 2(1) + 1 = 2 + 1 = 3 Step 5: Calculate output for x = 2. y = 2(2) + 1 = 4 + 1 = 5 Step 6: Create a table of values. |c|c| x & y \\ -2 & -3 \\ -1 & -1 \\ 0 & 1 \\ 1 & 3 \\ 2 & 5 \\ Step 7: Explain what m and c represent. In the equation y = 2x + 1: m = 2 represents the slope* of the line, indicating that for every unit increase in x, y increases by 2 units. c = 1 represents the y-intercept*, which is the value of y when x is 0. Table above, m is slope, c is y-intercept --- 19. Given the rule y = -2x + 5, determine the output when x = 3 and explain what this ordered pair represents in the context of the mapping. Step 1: Determine the output when x = 3. Substitute x = 3 into the rule: y = -2(3) + 5 y = -6 + 5 y = -1 Step 2: Explain what the ordered pair represents. The output when x = 3 is y = -1. This forms the ordered pair (3, -1). This ordered pair represents a specific point on the line defined by the mapping rule y = -2x + 5. It means that when the input is 3, the corresponding output is -1. The output when x = 3 is -1. The ordered pair (3, -1) represents a point on the line where an input of 3 yields an output of -1. --- 20. A taxi charges GH₵2.50 per kilometre plus a fixed charge of GH₵5. Write the mapping rule and calculate the cost for a 15 km journey. Step 1: Write the mapping rule. Let y be the total cost and x be the number of kilometres. The charge per kilometre is GH₵2.50, so m = 2.50. The fixed charge is GH₵5, so c = 5. The mapping rule is y = 2.50x + 5. Step 2: Calculate the cost for a 15 km journey. Substitute x = 15 into the rule: y = 2.50(15) + 5 y = 37.50 + 5 y = 42.50 The mapping rule is y = 2.50x + 5. The cost for a 15 km journey is GH₵42.50. --- Send me the next one 📸