Algebra - Graphs: identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically (Foundation tier)

Overview

A quadratic graph tells two versions of the same story: a curved picture and an algebraic rule. This topic teaches you to translate between them, identifying the roots, the intercepts and the turning point, then finding suitable roots exactly by factorising. Recognising those features helps candidates answer short graph-reading questions and longer questions combining a table of values, a drawing and an equation.

The central example is y = x² − 5x + 6. You will see why its roots are 2 and 3, why its y-intercept is (0, 6), and why its minimum lies between the roots rather than on the y-axis. Every step uses skills you already meet elsewhere: coordinates, substitution, negative numbers, expanding brackets and solving linear equations.

For Pearson Edexcel Foundation, the main specification reference is 1MA1 A11, supported by A18. Read turning points graphically and solve simple monic quadratics by factorising. Completing the square and the quadratic formula are Higher-tier methods, not requirements for this lesson. Typical command words include write down, estimate, factorise, solve, sketch and explain. 1 2

Key Concepts

1. Recognising a quadratic and its shape

A quadratic function can be written y = ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of x is two. Its graph is a parabola: one smooth curve, not a V and not a collection of straight line segments.

If a > 0, the parabola opens upwards and has a minimum. If a < 0, it opens downwards and has a maximum. The sign of the constant c does not decide the opening direction. For example, y = x² − 10 still opens upwards, whereas y = −x² + 10 opens downwards.

Why does the squared coefficient matter? For large positive or negative x, x² is positive and the ax² term dominates the linear and constant terms. A positive coefficient sends both ends upwards; a negative coefficient sends both ends downwards. The middle can move, but the ends retain that behaviour.

Cover and recall: Which opens downwards, y = x² − 4 or y = −x² + 4? Explain using the coefficient of x², not the constant.

2. Roots and x-intercepts: related, but not identical answers

A root is an x-value that makes the function zero. On a graph, a root is read where the curve meets the x-axis, because every point on that axis has y = 0. An x-intercept is the corresponding point, written as coordinates.

For y = x² − 5x + 6, the roots are x = 2 and x = 3. The x-intercepts are (2, 0) and (3, 0). If the question asks for roots, give the x-values. If it asks for coordinates of intersections, give both coordinate pairs. Read the actual wording rather than assuming the answer format.

This labelled graph is a learning aid. The gold point is the minimum; the coral points identify the roots; the dark point is the y-intercept. Cover the labels and explain what makes each feature different. The units on the two axes need not have the same physical spacing, so read each scale independently.

A graph can provide approximate roots even when the quadratic does not factorise into convenient integer brackets. Check the scale and estimate sensibly between gridlines. Do not write a long decimal that the printed graph cannot support.

3. Two roots, one repeated root, or no real roots

An upward-opening parabola can cross the x-axis twice, touch it once, or remain above it. A downward-opening parabola can likewise have two crossings, a single touch, or remain below it. Count where the entire relevant curve meets the axis, not just where it crosses through.

For y = x² − 4, the roots are −2 and 2. For y = x², the curve touches at (0, 0): x = 0 is one distinct real root, called a repeated root. For y = x² + 2, the minimum is above the x-axis, so there are no real roots. You do not need to calculate a discriminant or discuss complex numbers here.

Be cautious with a restricted graph window. An unseen part of the curve might meet the axis outside the printed interval. State conclusions supported by the graph and the information supplied.

Why question: Why does touching the axis count as a root? Because the function still equals zero there, even though the curve does not cross to the other side.

4. Finding the y-intercept by substitution

Every point on the y-axis has x = 0. Substitute zero into the equation, simplify, then write the result as (0, y).

For the main example, y = 0² − 5(0) + 6 = 6, so the y-intercept is (0, 6). For y = ax² + bx + c, the terms involving x vanish, leaving y = c. Therefore its y-intercept is (0, c).

Do not confuse (0, 6) with (6, 0). The first is on the y-axis; the second is on the x-axis. Nor should you assume that the y-intercept is the turning point: they coincide only when the turn is on the y-axis.

Cover and recall: For y = x² + 2x − 8, what is the y-intercept? Say which variable you set to zero before giving the answer.

5. Turning points and symmetry

The turning point, also called the vertex, is where the parabola changes from decreasing to increasing, or from increasing to decreasing. For a full upward-opening parabola, it is the lowest point. For a full downward-opening parabola, it is the highest point.

On Foundation, practise reading the turning point from a graph: locate the smooth turn, then read its x-coordinate and y-coordinate. If asked for coordinates, x = 2.5 alone is incomplete. The main example has turning point (2.5, −0.25), which is a minimum.

A parabola is symmetric about the vertical line through its turning point. If it has two real roots r₁ and r₂, those roots have the same height, zero, and are equally far from the symmetry line. Thus the turning point's x-coordinate is their midpoint: (r₁ + r₂)/2. This explains why the turn lies horizontally halfway between the two crossings.

Optional symmetry extension: For roots 2 and 3, the midpoint is (2 + 3)/2 = 2.5. Substituting gives y = 2.5² − 5(2.5) + 6 = 6.25 − 12.5 + 6 = −0.25. This checks the vertex without completing the square. Finding a vertex algebraically by completing the square is Higher-only; this guide's core skill remains graph reading. 2

The axis of symmetry is x = 2.5, a vertical line. It is not y = 2.5, and the midpoint does not imply that the turning point lies on the x-axis.

6. Deducing roots algebraically by factorising

For Foundation, focus on equations of the form x² + bx + c = 0 that can be solved by factorising. Pearson's teaching guidance limits Foundation quadratic equation solving to this monic form and this method. 2

To factorise x² + bx + c, find numbers p and q with p + q = b and pq = c. Then write (x + p)(x + q). Expanding explains the rule: the middle terms add to (p + q)x, while the constant is pq.

For x² − 5x + 6 = 0, choose −2 and −3: their product is 6 and their sum is −5. Hence (x − 2)(x − 3) = 0. By the zero-product rule, x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3. Check both in the original equation: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0.

For x² + 5x + 6 = 0, the brackets are (x + 2)(x + 3), but the roots are −2 and −3. Solve each bracket equation rather than copying its number. If c is negative, the bracket numbers have opposite signs; their sum still has to match b.

The zero-product rule requires a product equal to zero. From AB = 6, you cannot conclude A = 0 or B = 0. Also, factorising is not the final answer when the command word is solve.

Mathematical/Scientific Relationships

Formula-sheet status: The verified Pearson 2026 Foundation formula sheet does not supply the quadratic relationships below. Treat them as Must memorise, meaning understand and practise them until usable without the sheet. Pearson reports that formula sheets continue for the current specifications; always use the sheet for your own examination series. 3 4

• Must memorise: y = ax² + bx + c, a ≠ 0. Here x is the input, y the output, and a, b, c are coefficients/constants. It identifies a quadratic; a decides the opening direction.
• Must memorise: f(r) = 0 defines a root r. Its x-intercept is (r, 0). This is why setting y = 0 finds roots.
• Must memorise: y-intercept = (0, c) for y = ax² + bx + c. Use x = 0.
• Must memorise: (x + p)(x + q) = x² + (p + q)x + pq. Use p + q = b and pq = c to factorise a monic quadratic.
• Must memorise: AB = 0 implies A = 0 or B = 0. A and B stand for the factors, not necessarily single numbers.
• Must memorise: x² − k² = (x − k)(x + k). This difference-of-squares identity explains roots ±k, for example x² − 4 = 0 gives −2 and 2.
• Must memorise, optional extension: h = (r₁ + r₂)/2 and k = f(h), giving vertex (h, k) when two real roots are known. The axis of symmetry is x = h. Here k denotes the vertex height, not the k in the preceding identity.

Given on formula sheet: None of these Foundation quadratic-topic relationships. The quadratic formula and completing-square methods belong to Higher study and are deliberately not developed here.

Graph/Data Skills

For draw or plot, calculate every requested table entry, using brackets around negative x-values. For y = x² − 4x + 3 and x = −1, calculate (−1)² − 4(−1) + 3 = 1 + 4 + 3 = 8. Remember that (−2)² = 4, whereas −2² means −(2²) = −4.

For x-values −1, 0, 1, 2, 3, 4, 5, the corresponding y-values are 8, 3, 0, −1, 0, 3, 8. Plot small crosses, check paired coordinates, and draw a smooth curve through the points across the specified interval. The matching heights at x = 1 and 3, and at x = 0 and 4, help confirm symmetry about x = 2.

For sketch, show the essential shape and relevant features without pretending to have a precise plotted scale. Include coordinates when requested or useful to make the features unambiguous. Do not assume marks are automatically awarded for labels: credit given depends on the particular question and mark scheme.

Try before reading the worked solution: Use the graph below to estimate both roots of x² − 2x − 4 = 0 and read its turning point.

Pearson's teaching guidance gives a corresponding Foundation example with 2 marks for root estimates and 1 mark for turning-point coordinates. This graph is independently redrawn, not a scan of the exam figure. No official acceptance interval is claimed here. 2

Practical Applications

A ball's height against horizontal distance can be modelled by a downward-opening quadratic. Its maximum represents the greatest height; a root can represent where the ball reaches ground level. A negative root may lie outside the physical journey, so mathematical solutions must be interpreted within the stated domain.

Quadratics also arise from rectangular areas, because multiplying two changing lengths produces an x² term. That connects graphs to expanding brackets, not to a new graphing rule. Read the axes before attaching units: a height coordinate might be in metres while the horizontal coordinate is in seconds.

There is no required practical for this GCSE Mathematics topic. Pure algebraic coordinates have no automatic units. In contextual questions, keep units consistent before substitution: 250 cm = 2.5 m and 90 s = 1.5 min. Do not apply those conversions unless the question actually needs them.

Ten-minute Tutor Podcast

Listen with a pencil and answer the quiz aloud. The episode follows introduction, core concepts, exam tips, quick-fire recall, and summary. Afterwards, solve one different quadratic without replaying the explanation. Revisit missed rules tomorrow, then try again after several days.

References