OxfordAQA International GCSE Mathematics: Graphs of quadratic, cubic, reciprocal and other functions (9260) – Revision Notes

These revision notes condense references A12 and A16 of section 3.2.2 in the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). Core content is also assessable at Extension; items marked “Extension only” are the extra Extension content. The study guide explains each idea at greater length, with longer worked examples. Test yourself afterwards with the practice questions.

Related pages: the course hub, the printable checklist, the algebra manipulation revision notes and the free diagnostics.

Shapes at a glance

Equation	Picture in words	Remember
y = ax² + bx + c, a > 0	U-shape	Minimum; y-intercept c
y = ax² + bx + c, a < 0	∩-shape	Maximum; y-intercept c
y = x³	S-shape rising	Through the origin
y = −x³	S-shape falling	Through the origin
y = 1/x, x ≠ 0	Two branches	Quadrants 1 and 3; no contact with the axes
y = kˣ, k > 1 (Extension only)	Rising curve	Passes (0, 1); y > 0 always
y = kˣ, 0 < k < 1 (Extension only)	Falling curve	Passes (0, 1); y > 0 always
y = sin x, cos x (Extension only)	Wave	Range −1 to 1; period 360°
y = tan x (Extension only)	Repeating with breaks	Period 180°; breaks at 90°, 270° and so on

Quadratics: features and method

Method for features (Core)

1. y-intercept: set x = 0.
2. Roots: set y = 0 and factorise (or read the crossing points from a graph).
3. Axis of symmetry: x = (root₁ + root₂)/2.
4. Turning point: substitute the axis x-value into the equation.
5. Direction: sign of the x² coefficient.

Worked reminder

y = x² + 6x + 8 factorises as (x + 2)(x + 4), so the roots are −2 and −4. The axis is x = (−2 + −4)/2 = −3. Then y = 9 − 18 + 8 = −1, so the minimum is (−3, −1). The y-intercept is 8.

Number of roots

What the graph does	Roots
Crosses the x-axis twice	Two different roots
Touches the x-axis at the turning point	One repeated root, for example y = (x − 3)²
Stays wholly above or below the axis	No real roots

Solving f(x) = k graphically

Add the line y = k, parallel to the x-axis. The x-values where it meets the curve are the solutions. Two solutions are the same distance from the axis of symmetry, so if you know one you can find the other: x₂ = 2 × (axis) − x₁.

Completing the square (Extension only)

Form	Turning point	Type
y = (x − p)² + q	(p, q)	Minimum
y = q − (x − p)²	(p, q)	Maximum

Method for x² + bx + c: halve b, write (x + b/2)², subtract (b/2)², then add c.

Worked reminder: x² − 10x + 21 = (x − 5)² − 25 + 21 = (x − 5)² − 4. The minimum is (5, −4). Setting it to zero: (x − 5)² = 4, x − 5 = ±2, so the roots are 3 and 7.

Check: if q is positive on a minimum curve, or negative on a maximum curve, the curve never reaches the x-axis.

Cubics

• Factorised form shows the roots: y = (x − a)(x − b)(x − c) crosses at a, b and c.
• The y-intercept is found by setting x = 0 in the factorised form. For y = (x − a)(x − b)(x − c) it is −abc.
• Positive x³ coefficient: rises on the right and falls on the left. Negative: the reverse.
• Test one x-value between each pair of roots to see whether the curve is above or below the axis.
• y = x³ is rotationally symmetrical about the origin, so the points (2, 8) and (−2, −8) are both on it.

Worked reminder

Sketch y = (x + 1)(x − 3)(x − 5). The roots are −1, 3 and 5. At x = 0 the value is 1 × (−3) × (−5) = 15, so the y-intercept is 15. The x³ coefficient is positive, so the curve starts low on the left, crosses at −1, rises to a peak above the axis, crosses at 3, dips below the axis and crosses again at 5.

Reciprocal graph

• y = k/x: two separate branches, never touching either axis.
• Multiply: xy = k at every point.
• Positive k: quadrants 1 and 3. Negative k: quadrants 2 and 4.
• Symmetrical about y = x (positive k).

Sketch checklist

• Axes drawn and labelled, with the intercepts marked as numbers.
• Smooth curve with no straight segments.
• Correct end behaviour (check the sign of the highest power).
• The turning point marked, if the question asks for one.

Exponentials (Extension only)

• y = kˣ with k > 0.
• Always passes through (0, 1).
• Always above the x-axis.
• To find k, substitute a known point: if (2, 49) is on the curve, k² = 49, so k = 7.
• Negative powers are reciprocals: 3⁻² = 1/9.

Trigonometric graphs (Extension only)

Function	Period	Value at 0°	Features
sin x	360°	0	Maximum 1 at 90°, minimum −1 at 270°
cos x	360°	1	Minimum −1 at 180°, maximum 1 at 360°
tan x	180°	0	Breaks at 90° and 270°; zero at 0°, 180° and 360°

Finding all solutions in 0° to 360°

sin x = k   first angle a, second angle 180° − a
cos x = k   first angle a, second angle 360° − a
tan x = k   first angle a, second angle a + 180°
(add 360° for the next turn if the range is bigger)

Worked reminder: sin x = 0.8 between 0° and 180°. The inverse sine gives 53.1°, and the second is 180° − 53.1° = 126.9°.

For angles of any size, take away or add 360° (or 180° for tan) until the angle is in the 0° to 360° range, or use the repeating pattern of the graph.

Must-know distinctions

• Root vs intercept vs turning point. A root is on the x-axis. The y-intercept is on the y-axis. The turning point is where the curve changes direction. They are three separate things.
• y = x² vs y = x³. x² is even and has a turning point. x³ is odd and has none.
• y = 1/x vs y = kˣ. In 1/x the variable is the base, with two branches. In kˣ the variable is the power, with one curve through (0, 1).
• Period of sin and cos vs tan. 360° for sin and cos, 180° for tan.

Quick self-test

1. For y = x² − 14x + 40, find the roots and the turning point.
2. What does the graph of y = x² − 6x + 9 do at the x-axis?
3. (Extension only) Write y = x² + 4x + 7 in the form (x + p)² + q. Does the curve meet the x-axis?
4. y = 20/x. Find y when x = 4 and when x = −5.
5. (Extension only) y = 3ˣ. Find y when x = −2 and when x = 3.
6. (Extension only) Write down sin 150° and sin 450°.
7. (Extension only) Solve sin x = 0.8 for 0° ≤ x ≤ 180°. Give answers to 1 decimal place.
8. Write down the y-values of y = x³ at x = 2 and x = −2.
9. A cubic has equation y = x(x − 5)(x + 1). State its roots and find y when x = 2.
10. Find the axis of symmetry and turning point of y = 4x² − 24x.
11. (Extension only) State the period of y = tan x.

Answers

1. Factorise: (x − 4)(x − 10) = 0, so the roots are 4 and 10. Axis x = 7, and y = 49 − 98 + 40 = −9. Turning point (7, −9).
2. It is a perfect square, (x − 3)². The curve touches the axis at (3, 0), a repeated root.
3. (x + 2)² + 3, since (x + 2)² = x² + 4x + 4. The minimum is (−2, 3), above the axis, so it does not meet the x-axis.
4. y = 5 and y = −4.
5. y = 1/9 and y = 27.
6. sin 150° = 0.5 and sin 450° = sin 90° = 1.
7. x = 53.1° and x = 180° − 53.1° = 126.9°.
8. 8 and −8.
9. Roots 0, 5 and −1. At x = 2: 2 × (−3) × 3 = −18.
10. 4x(x − 6) = 0 gives roots 0 and 6. Axis x = 3, and y = 36 − 72 = −36. Turning point (3, −36).
11. 180°.

Where marks are usually lost

• Drawing a parabola as a V, or joining points with ruled segments.
• Taking the sign wrong in a completed square, so (x + 4)² + 1 is given a turning point at x = 4.
• Giving a turning point as a single number. A turning point is a coordinate pair.
• Reading a root from a curve that only touches the axis as if it crossed twice.
• Plotting y = 1/x with a line linking the two branches, or a curve running into the y-axis.
• Showing an exponential graph dropping below the x-axis, or leaving out the point (0, 1).
• Giving only one solution of a trig equation when the interval contains two.
• Leaving a sketch without labelled intercepts when the question asks for them.
• Writing a rounded decimal for a turning point when an exact form such as 3 ± √5 is easy to get.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1 (for exams May/June 2018 onwards), section 3.2.2, references A12 and A16, published by OxfordAQA.