A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Graphs of Functions and Proportional Relationships — mark scheme explained
The short answer
In AQA A-Level Mathematics, understanding and using graphs of functions is a crucial skill. This involves sketching various types of curves, interpreting algebraic solutions graphically, and solving equations by finding the intersection points of graphs. Additionally, you need to understand and use proportional relationships and their graphs. Sketching Curves 1.
The question
Sketch the graph of the polynomial function y = x 2 - 4x + 3 and find its roots.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For sketching graphs, marks are typically awarded for correctly identifying key features such as roots, y-intercepts, and asymptotes. For solving equations graphically, marks are given for accurately plotting the functions and finding intersection points.
What the command words demand
- Sketch
- Draw a graph of the function, including key features such as roots, y-intercepts, and asymptotes.
- Solve
- Find the solutions to an equation, either algebraically or graphically.
- Interpret
- Explain the meaning of a solution in the context of the problem or graph.
- Identify
- Recognize and label important points or features on a graph.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes per question involving sketching graphs or solving equations graphically to ensure you have enough time to identify key features and plot accurately.
- 1. Identify the roots by solving the equation x 2 - 4x + 3 = 0.0 marks
- 2. Factorize the quadratic: (x - 1)(x - 3) = 0, so the roots are x = 1 and x = 3.2 marks
- 3. Find the y-intercept by substituting x = 0 into the equation: y = 0 2 - 4(0) + 3 = 3.1 mark
- 4. Determine the end behavior: as x → ±∞, y → ∞ because the leading term is positive and has an even degree.0 marks
- 5. Plot the roots (1, 0) and (3, 0), the y-intercept (0, 3), and sketch the parabola opening upwards.2 marks
Final answer: The graph of y = x 2 - 4x + 3 is a parabola with roots at (1, 0) and (3, 0), a y-intercept at (0, 3), and it opens upwards.
Work through every step correctly and you earn all 5 marks.
Another worked example
Sketch the graph of y = |x - 2| and describe its key features.
- 1. Identify the vertex by setting x - 2 = 0, so the vertex is at (2, 0).1 mark
- 2. Plot the line y = x - 2 for x ≥ 2 and reflect it above the x-axis for x < 2.2 marks
- 3. The graph will be a V-shaped curve with its vertex at (2, 0).1 mark
Final answer: The graph of y = |x - 2| is a V-shaped curve with its vertex at (2, 0), and it consists of two linear segments: y = x - 2 for x ≥ 2 and y = -(x - 2) for x < 2.
Work through every step correctly and you earn all 4 marks.
Common mistakes
Forgetting to reflect the part of the graph below the x-axis when sketching the modulus function |ax + b|.
Why it happens: Students often forget that the modulus function reflects any part of the graph below the x-axis above it, creating a V-shaped curve.
Fix: Always check for and reflect the part of the graph below the x-axis when sketching the modulus function |ax + b|.
Incorrectly identifying the roots of a polynomial function by not solving the equation f(x) = 0.
Why it happens: Students sometimes skip the step of setting the polynomial equal to zero and solving for x, leading to incorrect roots.
Fix: Always set the polynomial equal to zero and solve for x to find the roots accurately.
Failing to identify vertical and horizontal asymptotes in rational functions.
Why it happens: Students may not recognize where the denominator is zero or how the degree of the numerator and denominator affects the horizontal asymptote.
Fix: Identify vertical asymptotes by finding where the denominator is zero and the numerator is non-zero. Determine horizontal asymptotes based on the degrees of the numerator and denominator.
Incorrectly interpreting intersection points when solving equations graphically.
Why it happens: Students may misread the x-coordinates of intersection points or fail to substitute back into one of the original equations to find the corresponding y-values.
Fix: Always read the x-coordinates of intersection points accurately and substitute them back into one of the original equations to find the y-values.
Forgetting that proportional relationships pass through the origin (0, 0).
Why it happens: Students may not recognize that a proportional relationship is a linear function of the form y = kx and must pass through the origin.
Fix: Always remember that proportional relationships are represented by straight lines passing through the origin (0, 0) with a slope of k.
Incorrectly determining the end behavior of polynomial functions.
Why it happens: Students may not consider the leading term and its sign when determining how the graph behaves as x → ±∞.
Fix: Always look at the leading term (the term with the highest degree) and its sign to determine the end behavior of the polynomial function.
Failing to plot key points when sketching graphs, such as roots, y-intercepts, and asymptotes.
Why it happens: Students may not take the time to identify and plot these important points, leading to inaccurate sketches.
Fix: Always identify and plot key points such as roots, y-intercepts, and asymptotes before sketching the graph.
Incorrectly solving quadratic equations by not factorizing or using the quadratic formula.
Why it happens: Students may skip steps or make algebraic errors when solving quadratic equations, leading to incorrect roots.
Fix: Always use appropriate methods such as factorization or the quadratic formula to solve quadratic equations accurately.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Sketch Quadratic Graph | Find the roots of a quadratic and sketch its parabola with key features labelled. | 5 |
| Sketch Modulus Graph | Sketch y = |x - 2| showing its V-shape, vertex, and axis intercepts. | 4 |
| Solve Equation Graphically | Plot two graphs and find their intersection points to solve the equation. | 6 |
| Sketch Reciprocal Graph | Sketch y = 1/x and describe its asymptotes, symmetry, and behaviour in each quadrant. | 5 |
| Total across these question types | 20 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.