AQA A-Level Mathematics: Graphs of Functions and Proportional Relationships — mark scheme explained

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

What each mark is really for — in plain English — and the wording trap that loses it.

• S1

  1. Identify the roots by solving the equation x 2 - 4x + 3 = 0.

• S2

  2. Factorize the quadratic: (x - 1)(x - 3) = 0, so the roots are x = 1 and x = 3.

• S3

  3. Find the y-intercept by substituting x = 0 into the equation: y = 0 2 - 4(0) + 3 = 3.

• S4

  4. Determine the end behavior: as x → ±∞, y → ∞ because the leading term is positive and has an even degree.

• S5

  5. Plot the roots (1, 0) and (3, 0), the y-intercept (0, 3), and sketch the parabola opening upwards.

Model answer

Worked through, with each step tagged to the mark it earns.

1. S1

   1. Identify the roots by solving the equation x 2 - 4x + 3 = 0.

2. S2

   2. Factorize the quadratic: (x - 1)(x - 3) = 0, so the roots are x = 1 and x = 3.

3. S3

   3. Find the y-intercept by substituting x = 0 into the equation: y = 0 2 - 4(0) + 3 = 3.

4. S4

   4. Determine the end behavior: as x → ±∞, y → ∞ because the leading term is positive and has an even degree.

5. S5

   5. Plot the roots (1, 0) and (3, 0), the y-intercept (0, 3), and sketch the parabola opening upwards.

6. 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.

Common mistakes

• Forgetting to reflect the part of the graph below the x-axis when sketching the modulus function |ax + b|. — 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. — 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. — 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. — 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). — 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. — 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. — 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. — Always use appropriate methods such as factorization or the quadratic formula to solve quadratic equations accurately.

Where the marks go

• Full worked solution (all marking points)5 marks

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