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

This study guide covers references A12 and A16 in section 3.2.2 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). It teaches the shapes and key features of the standard graphs, and how to read roots, intercepts and turning points from a quadratic. The specification says all content can be assessed on the Extension tier question papers, so Core content is also Extension content. Statements marked “Extension only” are the additional Extension content.

Use it with the revision notes and the practice questions. The course page is OxfordAQA IGCSE Mathematics and the printable list is the checklist. Not sure where you stand? Try a free 10-minute diagnostic. For the algebra underneath, see algebra notation and manipulation.

What this guide covers

Ref	What you must be able to do	Level
A12	Recognise, sketch and interpret graphs of linear and quadratic functions, simple cubic functions and the reciprocal function y = 1/x with x ≠ 0	Core
A12	Recognise, sketch and interpret y = kˣ for positive k	Extension only
A12	Recognise, sketch and interpret y = sin x, y = cos x and y = tan x for angles of any size, with arguments in degrees	Extension only
A16	Identify and interpret roots, intercepts and turning points of quadratics from a graph; deduce roots algebraically; use the symmetry of a quadratic	Core
A16	Deduce turning points by completing the square	Extension only

Straight-line graphs are covered elsewhere. Here a line is only a shape to recognise.

The standard shapes

Learn these on sight. Each one tells you something before you calculate anything.

Equation	Shape	What to look for
y = mx + c	Straight line	Gradient m; crosses the y-axis at c
y = x²	U-shaped parabola	Minimum at the origin, symmetrical about the y-axis
y = −x²	∩-shaped parabola	Maximum at the origin
y = x³	S-shaped curve	Passes through the origin, rises left to right, rotationally symmetrical about the origin
y = −x³	S-shape, falling	Falls from left to right
y = 1/x	Two separate branches	Never touches either axis; undefined at x = 0
y = kˣ (Extension only)	Curve with no turning point	Passes through (0, 1); never reaches the x-axis
y = sin x, cos x (Extension only)	Repeating wave	Between −1 and 1, repeats every 360°
y = tan x (Extension only)	Repeating, with breaks	Repeats every 180°; no maximum or minimum

The sign of the highest power decides the direction. A positive x² coefficient gives a U-shape. A negative one gives a ∩-shape. A positive x³ coefficient gives a rising S-shape.

Quadratic graphs

A quadratic function has the form y = ax² + bx + c with a ≠ 0. Its graph is a parabola. Four features matter.

• y-intercept. Put x = 0. The intercept is c.
• Roots. These are the x-values where y = 0, so the curve meets the x-axis. There can be two, one (the curve just touches the axis) or none.
• Axis of symmetry. A vertical line through the turning point. The two roots are the same distance either side of it, so the axis is x = (root₁ + root₂)/2.
• Turning point. The minimum (U-shape) or maximum (∩-shape). It lies on the axis of symmetry.

Worked example 1: features from the equation

Find the roots, y-intercept, axis of symmetry and turning point of y = x² − 6x − 7.

Roots:        x² − 6x − 7 = 0
              (x − 7)(x + 1) = 0
              x = 7 or x = −1
y-intercept:  x = 0 gives y = −7
Axis:         x = (7 + (−1))/2 = 3
Turning point: x = 3 gives y = 9 − 18 − 7 = −16

The coefficient of x² is positive, so the turning point is a minimum at (3, −16). A sketch is a U-shape through (−1, 0), (0, −7) and (7, 0), with its lowest point at (3, −16).

Notice the symmetry in the intercepts. The point (0, −7) has a partner at x = 6, because 6 is as far right of the axis as 0 is left of it. So (6, −7) is also on the curve.

Reading a quadratic graph

Questions often give a graph, or a table of values, rather than an equation.

• Roots are where the curve crosses the x-axis.
• To solve f(x) = k, add a line parallel to the x-axis at height k. Its crossing points with the curve give the solutions.
• The turning point is the highest or lowest point. Its x-value is halfway between a pair of points with equal y-values.

Worked example 2: using symmetry

A parabola passes through (2, 9) and (8, 9), and its roots are at x = −3 and x = 13. Find the axis of symmetry and state the x-coordinate of the turning point.

Points with equal y-values are the same distance from the axis, so the axis is x = (2 + 8)/2 = 5. The roots give the same answer: (−3 + 13)/2 = 5. The turning point has x-coordinate 5.

Completing the square (Extension only)

Completing the square gives the turning point directly. A curve written as y = (x − p)² + q has its turning point at the point (p, q). The squared bracket is never negative, so its smallest value is 0, when x = p. If the curve is y = q − (x − p)², the bracket is subtracted, so (p, q) is a maximum.

Worked example 3: a minimum

Complete the square for y = x² − 8x + 9 and give its turning point.

x² − 8x + 9 = (x − 4)² − 16 + 9
            = (x − 4)² − 7

The turning point is (4, −7), a minimum. The roots can be found from the same form: (x − 4)² = 7, so x = 4 ± √7, which is 1.35 or 6.65 to 2 decimal places.

Worked example 4: a maximum

Complete the square for y = 3 + 4x − x², writing it as q − (x − p)², and give the turning point.

3 + 4x − x² = −(x² − 4x) + 3
            = −[(x − 2)² − 4] + 3
            = 7 − (x − 2)²

The turning point is (2, 7), a maximum, so y ≤ 7 for every x.

Simple cubic graphs

A cubic has x³ as its highest power. The A12 content is simple cubics, such as y = x³, y = −x³ and cubics already in factorised form. A cubic can cross the x-axis up to three times.

Worked example 5: sketching a factorised cubic

Sketch y = x(x − 2)(x + 4).

Roots:       x = 0, x = 2, x = −4
y-intercept: 0
Sign check:  x = 1 gives 1 × (−1) × 5 = −5  (below the axis)
             x = −1 gives (−1)(−3)(3) = 9   (above the axis)
Ends:        the x³ coefficient is +1, so the curve rises on the far right
             and falls on the far left

Starting low on the left, the curve crosses the axis at −4 and peaks above it between −4 and 0. It crosses at 0, dips below the axis until 2, then crosses again and keeps rising. Label the three roots on the axis. Expanded, the equation is y = x³ + 2x² − 8x, whose constant term is 0, matching the intercept.

The reciprocal graph y = 1/x

For y = 1/x, x cannot be 0. The graph has two branches, one in the top-right quadrant (x > 0, y > 0) and one in the bottom-left (x < 0, y < 0). As x gets larger the curve gets closer to the x-axis but never meets it. As x gets closer to 0 the curve climbs or falls steeply but never meets the y-axis. The graph is symmetrical about the line y = x.

For y = k/x with k > 0, the shape is the same and each y-value is multiplied by k.

Worked example 6: y = 6/x

x:   −6   −3   −2   −1    1    2    3    6
y:   −1   −2   −3   −6    6    3    2    1

Plot the points and join each branch with a smooth curve. Do not join the two branches, and do not let either curve touch an axis. If k is negative, the branches move to the other pair of quadrants.

Exponential graphs y = kˣ (Extension only)

In y = kˣ, the variable is the power. Take k > 0 and k ≠ 1 (when k = 1 the graph is just the horizontal line y = 1).

• Every such graph passes through (0, 1), because k⁰ = 1.
• If k > 1 the curve rises quickly to the right and gets close to the negative x-axis on the left.
• If 0 < k < 1 the curve falls, flattening towards the positive x-axis.
• y is always positive. The curve never reaches the x-axis.

x:         −2     −1     0     1     2     3
y = 2ˣ:   0.25   0.5    1     2     4     8
y = (½)ˣ:   4     2     1    0.5   0.25  0.125

The two curves are mirror images in the y-axis.

Worked example 7: finding k

The curve y = kˣ passes through (3, 125). Find k and the value of y when x = −1.

k³ = 125, so k = 5
x = −1:  y = 5⁻¹ = 1/5 = 0.2

Trigonometric graphs (Extension only)

The arguments are in degrees, and the angles can be any size, including negative angles and angles above 360°.

Function	Period	Range	Key points
y = sin x	360°	−1 to 1	Starts at 0, maximum at 90°, zero at 180°, minimum at 270°
y = cos x	360°	−1 to 1	Starts at 1, zero at 90°, minimum at 180°, maximum at 360°
y = tan x	180°	All values	Zero at 0° and 180°; breaks at 90° and 270°

The cosine curve is the sine curve shifted 90° to the left. Because each pattern repeats, sin 390° = sin 30° = 0.5. Negative angles run the graph backwards: sin(−30°) = −0.5.

Worked example 8: solving sin x = k and cos x = k on the graph

Solve cos x = −0.4 for 0° ≤ x ≤ 360°.

The inverse cosine gives x = 113.6° (1 decimal place). The cosine curve is symmetrical about 180°, so the second solution is 360° − 113.6° = 246.4°. Both solutions are where the horizontal line y = −0.4 meets the curve.

Worked example 9: tan x = 2

Solve tan x = 2 for 0° ≤ x ≤ 360°. The first solution is x = 63.4°. The tan curve repeats every 180°, so the next is 63.4° + 180° = 243.4°.

Common errors

• Drawing a parabola with a pointed bottom, or joining plotted points with straight segments. Draw a smooth curve.
• Giving the turning point of y = (x − 4)² − 7 as (−4, −7). The sign inside the bracket flips: it is (4, −7).
• Finding the turning point as the midpoint of the y-intercept and a root. Use the midpoint of the two roots.
• Joining the two branches of y = 1/x, or letting them touch the axes.
• Drawing y = kˣ so that it goes below the x-axis, or missing (0, 1).
• Starting the sine curve at 1 or the cosine curve at 0.
• Giving only the first solution of a trig equation in the interval.

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. Check oxfordaqa.com for the latest version.