Functions and modelling

Paper labels are an unofficial, independently authored grouping, applied only where a question's own format genuinely matches a real paper convention (such as IB Mathematics AA's non-calculator/calculator split). They do not reproduce any exam board's real paper numbering or mark allocation, and uncertain questions are labelled general practice instead.

• 1A gym charges a joining fee plus a fixed cost per visit. A member who makes 2 visits pays a total of $15, and a member who makes 6 visits pays a total of $35. Assuming the relationship is linear, find the equation for the total cost C in terms of the number of visits n, and interpret the gradient and the C-intercept in context.Paper 1 styleMedium4 marksCalculator+
• 2Marking analysis: A learner attempts the following task: “A gym charges a joining fee plus a fixed cost per visit. A member who makes 2 visits pays a total of $15, and a member who makes 6 visits pays a total of $35. Assuming the relationship is linear, find the equation for the total cost C in terms of the number of visits n, and interpret the gradient and the C-intercept in context.” Their response addresses only this point: “Calculates the gradient as (35 − 15)/(6 − 2) = 5.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksCalculator+
• 3The height of a ball thrown in the air is modelled by h(t) = −5t² + 20t + 1, where h is in metres and t is in seconds. (a) Find the time at which the ball reaches its maximum height. (b) Find the maximum height reached.Paper 1 styleEasy3 marksCalculator+
• 4Marking analysis: A learner attempts the following task: “The height of a ball thrown in the air is modelled by h(t) = −5t² + 20t + 1, where h is in metres and t is in seconds. (a) Find the time at which the ball reaches its maximum height. (b) Find the maximum height reached.” Their response addresses only this point: “Uses the vertex formula t = −b/(2a) with a = −5, b = 20 to obtain t = 2.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksCalculator+
• 5A bacteria population grows exponentially according to P(t) = P0·e^(kt). The initial population is 200, and after 5 hours the population is 600. (a) Find the value of k. (b) Use your model to predict the population after 10 hours.Paper 1 styleMedium4 marksCalculator+
• 6Marking analysis: A learner attempts the following task: “A bacteria population grows exponentially according to P(t) = P0·e^(kt). The initial population is 200, and after 5 hours the population is 600. (a) Find the value of k. (b) Use your model to predict the population after 10 hours.” Their response addresses only this point: “Substitutes P(5) = 600 and P0 = 200 into 600 = 200e^(5k).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksCalculator+
• 7A radioactive substance has a half-life of 8 days. A sample initially has a mass of 50 grams. (a) Find the decay constant k in the model m(t) = 50e^(−kt). (b) Find the mass remaining after 20 days.Paper 1 styleMedium4 marksCalculator+
• 8Marking analysis: A learner attempts the following task: “A radioactive substance has a half-life of 8 days. A sample initially has a mass of 50 grams. (a) Find the decay constant k in the model m(t) = 50e^(−kt). (b) Find the mass remaining after 20 days.” Their response addresses only this point: “Uses the half-life condition m(8) = 25 to set up 25 = 50e^(−8k).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksCalculator+
• 9The sound intensity level L in decibels is modelled by L = 10 log10(I/I0), where I is the intensity in W/m² and I0 = 1 × 10⁻¹² W/m² is the reference intensity. Calculate the sound intensity level of a sound with intensity I = 1 × 10⁻⁶ W/m².Paper 1 styleEasy3 marksCalculator+
• 10Marking analysis: A learner attempts the following task: “The sound intensity level L in decibels is modelled by L = 10 log10(I/I0), where I is the intensity in W/m² and I0 = 1 × 10⁻¹² W/m² is the reference intensity. Calculate the sound intensity level of a sound with intensity I = 1 × 10⁻⁶ W/m².” Their response addresses only this point: “Substitutes I = 1×10⁻⁶ and I0 = 1×10⁻¹² into the formula.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksCalculator+
• 11The cost y of buying x identical notebooks varies directly with x. If 6 notebooks cost $24, find the constant of variation, and use it to find the cost of 10 notebooks.Paper 1 styleEasy3 marksCalculator+
• 12Marking analysis: A learner attempts the following task: “The cost y of buying x identical notebooks varies directly with x. If 6 notebooks cost $24, find the constant of variation, and use it to find the cost of 10 notebooks.” Their response addresses only this point: “Sets up y = kx and substitutes y = 24, x = 6 to obtain k = 4.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksCalculator+
• 13The time y taken to complete a job varies inversely with the number of workers x. If 8 workers take 5 hours, find the constant of variation, and use it to find how long 20 workers would take.Paper 1 styleEasy3 marksCalculator+
• 14Marking analysis: A learner attempts the following task: “The time y taken to complete a job varies inversely with the number of workers x. If 8 workers take 5 hours, find the constant of variation, and use it to find how long 20 workers would take.” Their response addresses only this point: “Sets up y = k/x and substitutes y = 5, x = 8 to obtain k = 40.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksCalculator+
• 15A parking garage charges according to the piecewise function C(h) = 5 if 0 < h ≤ 1, C(h) = 5 + 3(h − 1) if h > 1, where h is the number of hours parked and C is the cost in dollars. Calculate the cost of parking for (a) 1 hour, and (b) 4 hours.Paper 1 styleEasy3 marksCalculator+
• 16Marking analysis: A learner attempts the following task: “A parking garage charges according to the piecewise function C(h) = 5 if 0 < h ≤ 1, C(h) = 5 + 3(h − 1) if h > 1, where h is the number of hours parked and C is the cost in dollars. Calculate the cost of parking for (a) 1 hour, and (b) 4 hours.” Their response addresses only this point: “Identifies that h = 1 falls in the first piece, giving a cost of $5.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksCalculator+
• 17Solve the equation 3e^(2x) = 45 for x, giving your answer to 3 significant figures.Paper 1 styleEasy3 marksCalculator+
• 18Marking analysis: A learner attempts the following task: “Solve the equation 3e^(2x) = 45 for x, giving your answer to 3 significant figures.” Their response addresses only this point: “Divides both sides by 3 to obtain e^(2x) = 15.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksCalculator+
• 19Solve the equation log2(x) + log2(x − 2) = 3 for x.Paper 1 styleMedium4 marksCalculator+
• 20Marking analysis: A learner attempts the following task: “Solve the equation log2(x) + log2(x − 2) = 3 for x.” Their response addresses only this point: “Uses the law of logarithms to combine the left side: log2[x(x − 2)] = 3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksCalculator+
• 21Given f(x) = 2x + 1 and g(x) = x² − 3, find (a) f(g(2)) and (b) g(f(2)).Paper 1 styleEasy3 marksNo calculator+
• 22Marking analysis: A learner attempts the following task: “Given f(x) = 2x + 1 and g(x) = x² − 3, find (a) f(g(2)) and (b) g(f(2)).” Their response addresses only this point: “Calculates g(2) = 2² − 3 = 1, then f(1) = 2(1) + 1 = 3, giving f(g(2)) = 3.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy3 marksNo calculator+
• 23Find the inverse function f⁻¹(x) for f(x) = (3x − 2)/5, and verify your answer by finding f⁻¹(f(4)).Paper 1 styleMedium4 marksNo calculator+
• 24Marking analysis: A learner attempts the following task: “Find the inverse function f⁻¹(x) for f(x) = (3x − 2)/5, and verify your answer by finding f⁻¹(f(4)).” Their response addresses only this point: “Sets y = (3x − 2)/5 and swaps x and y to begin solving for the inverse.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksNo calculator+
• 25A model predicts the number of visitors to a museum as V(x) = 100√(20 − x), where x is the entry price in dollars. Explain why the domain of this model should be restricted to 0 ≤ x ≤ 20 in this real-world context.Paper 1 styleEasy2 marksNo calculator+
• 26Marking analysis: A learner attempts the following task: “A model predicts the number of visitors to a museum as V(x) = 100√(20 − x), where x is the entry price in dollars. Explain why the domain of this model should be restricted to 0 ≤ x ≤ 20 in this real-world context.” Their response addresses only this point: “Explains that x cannot be negative, since an entry price cannot be below zero, giving the lower bound x ≥ 0.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy2 marksNo calculator+
• 27The graph of y = x² is transformed to the graph of y = (x − 3)² + 2. Describe the two transformations applied, and state the coordinates of the vertex of the transformed graph.Paper 1 styleEasy2 marksNo calculator+
• 28Marking analysis: A learner attempts the following task: “The graph of y = x² is transformed to the graph of y = (x − 3)² + 2. Describe the two transformations applied, and state the coordinates of the vertex of the transformed graph.” Their response addresses only this point: “Describes a horizontal translation of 3 units to the right, and a vertical translation of 2 units up.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisEasy2 marksNo calculator+
• 29The height of the tide in a harbour, h metres, t hours after midnight, is modelled by h(t) = 5 + 3sin(πt/6). (a) State the amplitude and the period of this model. (b) Calculate the height of the tide at t = 3 hours.Paper 2 styleMedium4 marksCalculator+
• 30Marking analysis: A learner attempts the following task: “The height of the tide in a harbour, h metres, t hours after midnight, is modelled by h(t) = 5 + 3sin(πt/6). (a) State the amplitude and the period of this model. (b) Calculate the height of the tide at t = 3 hours.” Their response addresses only this point: “States the amplitude as 3 metres.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksCalculator+
• 31Find the coordinates of the points of intersection of the line y = 2x + 1 and the curve y = x² − 4x + 5.Paper 2 styleMedium5 marksCalculator+
• 32Marking analysis: A learner attempts the following task: “Find the coordinates of the points of intersection of the line y = 2x + 1 and the curve y = x² − 4x + 5.” Their response addresses only this point: “Sets the two expressions for y equal: 2x + 1 = x² − 4x + 5.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium5 marksCalculator+