Mathematics AA: Standard Level
Functions — Topic 2
- 1.
The function f(x) = x² − 4x + 7 is defined for all real x. Write f(x) in completed-square form. Hence state the coordinates of the vertex and the minimum value of f.
[3 marks] · no calculator - 2.
Marking analysis: A learner attempts the following task: “The function f(x) = x² − 4x + 7 is defined for all real x. Write f(x) in completed-square form. Hence state the coordinates of the vertex and the minimum value of f.” Their response addresses only this point: “Correctly obtains f(x) = (x − 2)² + 3.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 3.
A population is modelled by P(t) = 1200(1.08)^t, where t is the number of complete years after the start of 2026. Find the first value of t for which the population exceeds 2000. Show your method.
[4 marks] - 4.
Marking analysis: A learner attempts the following task: “A population is modelled by P(t) = 1200(1.08)^t, where t is the number of complete years after the start of 2026. Find the first value of t for which the population exceeds 2000. Show your method.” Their response addresses only this point: “Sets up 1200(1.08)^t > 2000 or the corresponding equality.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 5.
Solve the inequality x² − 5x + 6 ≤ 0. Give your answer using interval notation.
[3 marks] · no calculator - 6.
Marking analysis: A learner attempts the following task: “Solve the inequality x² − 5x + 6 ≤ 0. Give your answer using interval notation.” Their response addresses only this point: “Factorises to (x − 2)(x − 3) ≤ 0.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 7.
The function f(x) = 3x − 2 is defined for all real x. Find f⁻¹(x) and state its domain.
[3 marks] · no calculator - 8.
Marking analysis: A learner attempts the following task: “The function f(x) = 3x − 2 is defined for all real x. Find f⁻¹(x) and state its domain.” Their response addresses only this point: “Sets y = 3x − 2 and swaps x and y (or equivalent method).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 9.
Let f(x) = 2x + 1 and g(x) = x². Find an expression for the composite function (f ∘ g)(x) and evaluate (f ∘ g)(3).
[3 marks] · no calculator - 10.
Marking analysis: A learner attempts the following task: “Let f(x) = 2x + 1 and g(x) = x². Find an expression for the composite function (f ∘ g)(x) and evaluate (f ∘ g)(3).” Their response addresses only this point: “Substitutes g(x) into f to obtain f(g(x)) = 2x² + 1.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 11.
Solve the equation log₂(x) + log₂(x − 2) = 3 for x, stating why one algebraic solution must be rejected.
[5 marks] - 12.
Marking analysis: A learner attempts the following task: “Solve the equation log₂(x) + log₂(x − 2) = 3 for x, stating why one algebraic solution must be rejected.” Their response addresses only this point: “Uses the log law log₂(x) + log₂(x − 2) = log₂[x(x − 2)].” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 13.
A bacteria culture grows according to N(t) = 500(2)^(t/3), where t is measured in hours. Find the initial population and calculate how long it takes for the population to reach 4000.
[4 marks] - 14.
Marking analysis: A learner attempts the following task: “A bacteria culture grows according to N(t) = 500(2)^(t/3), where t is measured in hours. Find the initial population and calculate how long it takes for the population to reach 4000.” Their response addresses only this point: “States the initial population N(0) = 500.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 15.
The graph of y = f(x) is transformed to give the graph of y = f(x − 2) + 3. Describe fully the geometric transformation applied to the graph of f.
[2 marks] · no calculator - 16.
Marking analysis: A learner attempts the following task: “The graph of y = f(x) is transformed to give the graph of y = f(x − 2) + 3. Describe fully the geometric transformation applied to the graph of f.” Their response addresses only this point: “Identifies a horizontal translation of 2 units in the positive x-direction.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator - 17.
Solve the equation e^(2x) = 15, giving your answer to three significant figures.
[3 marks] - 18.
Marking analysis: A learner attempts the following task: “Solve the equation e^(2x) = 15, giving your answer to three significant figures.” Their response addresses only this point: “Takes the natural logarithm of both sides: 2x = ln(15).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 19.
Find the largest possible domain of f(x) = √(x − 3), and state the corresponding range.
[3 marks] · no calculator - 20.
Marking analysis: A learner attempts the following task: “Find the largest possible domain of f(x) = √(x − 3), and state the corresponding range.” Their response addresses only this point: “Requires the expression under the root to be non-negative: x − 3 ≥ 0.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 21.
The quadratic equation kx² + 4x + 1 = 0 has two equal real roots. Find the value of k.
[3 marks] - 22.
Marking analysis: A learner attempts the following task: “The quadratic equation kx² + 4x + 1 = 0 has two equal real roots. Find the value of k.” Their response addresses only this point: “States that for equal roots, the discriminant b² − 4ac = 0.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 23.
A parabola has vertex (1, −4) and passes through the point (3, 4). Find the equation of the parabola in the form y = a(x − h)² + k.
[4 marks] - 24.
Marking analysis: A learner attempts the following task: “A parabola has vertex (1, −4) and passes through the point (3, 4). Find the equation of the parabola in the form y = a(x − h)² + k.” Their response addresses only this point: “Writes the vertex form using the given vertex: y = a(x − 1)² − 4.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 25.
A radioactive sample decays according to the model M(t) = 80e^(−0.05t), where M is the mass in grams and t is the time in years. Find the initial mass, and calculate the time taken for the mass to decay to 20 grams.
[4 marks] - 26.
Marking analysis: A learner attempts the following task: “A radioactive sample decays according to the model M(t) = 80e^(−0.05t), where M is the mass in grams and t is the time in years. Find the initial mass, and calculate the time taken for the mass to decay to 20 grams.” Their response addresses only this point: “States the initial mass M(0) = 80 g.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 27.
Solve the equation 2ln(x) − ln(x − 1) = ln(4) for x, given that x > 1.
[5 marks] - 28.
Marking analysis: A learner attempts the following task: “Solve the equation 2ln(x) − ln(x − 1) = ln(4) for x, given that x > 1.” Their response addresses only this point: “Uses the power law to rewrite 2ln(x) as ln(x²).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 29.
State the domain of the function f(x) = 1/(x − 3), and write the equations of its vertical and horizontal asymptotes.
[3 marks] · no calculator - 30.
Marking analysis: A learner attempts the following task: “State the domain of the function f(x) = 1/(x − 3), and write the equations of its vertical and horizontal asymptotes.” Their response addresses only this point: “States the domain as all real x except x = 3 (x ≠ 3), since division by zero is undefined.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 31.
f(x) = 3x - 5 for x >= 2. Find its inverse and state the domain of the inverse.
[3 marks] · no calculator - 32.
Marking analysis: A learner attempts the following task: “f(x) = 3x - 5 for x >= 2. Find its inverse and state the domain of the inverse.” Their response addresses only this point: “The range of f is y >= 1.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator