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Quadratic and other function graphs - Mathematics Year 10

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Introduction

A quadratic can have two, one or no real horizontal-axis crossings. Connect its equation, table, symmetry and turning point rather than assuming every U-shaped graph crosses twice.

Sketch three quadratics with labelled roots and turning points. A separate Higher-tier reciprocal extension examines its forbidden input and two branches. Check algebra and actual plotted values before accepting a visual guess.

Sketch the source quadratic from equivalent forms

For the sourcey=x²−6 x+8, factorisation givesy=(x −2)(x −4), so the real roots are 2 and 4. Completing the square gives $y = (x - 3)^{2} - 1.$ The square is nonnegative, so the minimum output is −1 whenx=3. Thus the turning point is(3, −1), and the symmetry line isx=3.

These are two equivalent forms of one function. The factored form exposes roots; the completed-square form exposes the turning point. Re-expanding either expression recoversx²−6 x+8, which is a stronger check than relying on the appearance of a sketch.

First complete quadratic sketch. Calculate the following values before marking them. Equal distances fromx=3 give equal outputs: inputs 2 and 4 both give 0, 1 and 5 both give 3, and 0 and 6 both give 8.

x	y=x²−6 x+8
0	8
1	3
2	0
3	−1
4	0
5	3
6	8

y
 8 ●                 ●
 7 |
 6 |
 5 |
 4 |
 3 |  ●           ●
 2 |
 1 |
 0 +-----●-----●------→ x
-1 |        ●
   0  1  2  3  4  5  6

Text alternative: sample points(0, 8), (1, 3), (2, 0), (3, −1), (4, 0), (5, 3), (6, 8), with turning point(3, −1), roots 2 and 4 and symmetry linex=3. The schematic marks points on labelled axes; draw a smooth parabola through the verified points rather than joining them into a polygon.

The y-intercept is(0, 8), not one of the roots. A root is an input making output 0; a y-intercept uses input 0. Confusing these roles can turn the correct constant 8 into an incorrect alleged root. Substitutingx=8 in the original gives 24, not 0.

For a check between sampled integer positions, x=2.5 gives.25−1=−.75 in the completed-square form. The graph is below the axis between its two roots. The symmetry relationf(3−d)=f(3+d) follows because both sides contain the same squared distance d². A finite table illustrates this; the algebra establishes it for every reald.

Keep the source domain as all realx. The graphical image is a representation, not a physical measurement whose rounding defines the roots. Algebra gives exact roots and turning point; an approximate hand sketch should be labelled consistently with those exact results.

Compare three quadratic root-count cases

Second complete quadratic sketch: one repeated real root. For y=(x −1)², the square is never negative. Its minimum 0 occurs only atx=1, so the turning point is(1, 0), the symmetry line isx=1, and the one distinct real root is 1. The graph touches the horizontal axis there rather than crossing through it.

x	y=(x −1)²
−1	4
0	1
1	0
2	1
3	4

y
 4 ●  |        ●
 3    |
 2    |
 1    ●     ●
 0 ---+--●--------→ x
  -1  0  1  2  3

Text alternative: points(−1, 4), (0, 1), (1, 0), (2, 1), (3, 4). The y-intercept is(0, 1), while the root and minimum are at(1, 0). Substituting negative inputs means squaring the whole x −1 expression; at −1, it is(−2)²=4.

Third complete quadratic sketch: no real roots. For y=x²+1, the minimum output is 1 atx=0. The turning point is(0, 1), the symmetry line isx=0, and the y-intercept is also(0, 1). Because a real square cannot be −1, the equationx²+1=0 has no real solution.

x	y=x²+1
−2	5
−1	2
0	1
1	2
2	5

y
 5 ●     |     ●
 4       |
 3       |
 2    ●  |  ●
 1       ●
 0 ------+--------→ x
  -2 -1  0  1  2

Text alternative: points(−2, 5), (−1, 2), (0, 1), (1, 2), (2, 5). All outputs are above 0. These sampled points support the sketch, but the nonnegative-square argument proves that no real root is missing between them or beyond the displayed window.

Compare all three cases: the first has minimum −1 and two roots, the second minimum 0 and one distinct repeated root, the third minimum 1 and no real roots. The count depends on the actual function, not on a rule that every quadratic crosses the axis twice. These three upward-opening examples do not claim that every quadratic minimum is positive or that every parabola opens upward.

The actual original Corbettmaths tutorial plots the different functiony=x²+2 x −1 using a value table and coordinate axes. Follow the handling of negative inputs and smooth plotting. Our three equations and their root counts must be checked from their own data, not copied from that tutorial. Original: corbettmaths, Drawing quadratics - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Higher-tier reciprocal domain and graph-family transfer

Higher-tier extension: the reciprocal family. For $y = \frac{1}{x}$ , x=0 is forbidden because division by 0 is undefined. Positive inputs give positive outputs and negative inputs give negative outputs, so the graph has two branches in the first and third quadrants. It has neither an x-intercept nor a y-intercept.

x	y=1/x
−4	−0.25
−2	−0.5
−1	−1
−0.5	−2
0.5	2
1	1
2	0.5
4	0.25

Do not connect the negative and positive branches acrossx=0. As inputs approach 0 from the positive side, outputs grow without bound; from the negative side, they become unboundedly negative. For larger magnitudes ofx, the output approaches 0 but never equals 0. The axes are asymptotes, not extra parts of the graph.

The reciprocal table has an independent invariantxy=1. For example, (−.5)×(−2)=1 and 4×.25=1. The input domain and output range both exclude 0. This differs from a quadratic, which is defined at every real input but may have a minimum restricting its outputs.

For further graph-family recognition, small examples ofy=x³ are −8, −1, 0, 1, 8 atx −2, −1, 0, 1, 2. The cubic is not symmetric about the y-axis; its sign changes with the input. This is a separate progression example, not a requirement to apply the quadratic turning-point method to every family.

For unfamiliar quadratic transfer, y=x²−4 x+3=(x −1)(x −3)=(x −2)²−1. Roots are 1 and 3, turning point(2, −1), and symmetry linex=2. Atx=0, the output 3 is the y-intercept. Re-expansion of both alternate forms checks that these features belong to the same expression.

First-error repair. A learner claims the source turning point is atx=−3 because its square form containsx −3. The square is smallest whenx −3=0, which givesx=3. Another assigns a reciprocal point(0, 0) to connect the branches; that input is outside the domain, and the product checkxy=1 would fail.

Keep three labelled quadratic sketches with exact features and checked tables, the reciprocal domain statement, an unfamiliar transfer and a first-error repair. Distinguish values justified by algebra from approximate drawing positions. An actual later independent sketch must be completed before it can be recorded as new evidence.

Interactive Tasks

Quiz: Test Your Knowledge

Source x²−6 x+8 has real roots? (2 and 4) (!-2 and -4) (!3 only) (!No real roots)

Source quadratic turning-point x coordinate? (3) (!-3) (!2) (!4)

Source turning-point output? (-1) (!1) (!0) (!8)

Source symmetry line? (x=3) (!y=3) (!x=−3) (!y=−1)

Square of x −1 has how many distinct real roots? (One; root 1) (!Two; roots 1 and -1) (!None) (!Three)

Minimum of x²+1 over real inputs? (1) (!0) (!-1) (!No minimum)

How many real roots does x²+1 have? (None) (!One) (!Two) (!Every real input)

Higher tier: which reciprocal input is forbidden? (0) (!1) (!-1) (!2)

Higher tier: reciprocal output at −0.5? (-2) (!2) (!-0.5) (!0.5)

New x²−4 x+3 has turning point? (x=2, y=−1) (!x=−2, y=−1) (!x=1, y=0) (!x=3, y=0)

Memory Game

Root	Input with output zero
Turning point	Vertex where the quadratic changes direction
Symmetry	Equal outputs at equal distances from the symmetry line
Domain	Permitted input values
Asymptote	Line approached without becoming a graph branch
Reciprocal	One divided by a nonzero input

Drag and Drop

2 and 4	Source real root pair
x=3, y=−1	Source turning point
1 only	Distinct root of the second quadratic
No real roots	Root count for the third quadratic
x≠0	Higher-tier reciprocal domain restriction

Crossword Puzzle

QUADRATIC	Polynomial function with highest input power two
VERTEX	Another name for a parabola turning point
SYMMETRY	Matching across a stated line
RECIPROCAL	Function given by one divided by input
DOMAIN	Permitted input set
ASYMPTOTE	Line approached by the curve

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Read the source table: Find the two zero outputs and the minimum.
Name the symmetry line: Explain equal outputs at 2 and 4.
Distinguish intercepts: Compare input 0 with output 0.
Read the three sketches: Identify each labelled turning point.

Standard

Complete three quadratic sketches: Use verified tables, symmetry, roots and turning points.
Prove source features: Compare factored and completed-square forms.
Explain root counts: Use the minimum and nonnegative-square reasoning.
Repair a shifted-square error: Explain why x −3 gives symmetry at positive 3.

Advanced

Higher tier: state reciprocal restrictions: Give domain, branches, nonzero range and asymptotes.
Higher tier: check the reciprocal table: Use the invariant xy=1 without crossing the missing input.
Transfer to a new quadratic: Find roots, vertex and symmetry forx²−4 x+3.
Compare graph families: Explain why a cubic or reciprocal needs different structural reasoning.

Learning Check

Source quadratic: State exact roots, vertex and symmetry.
Second quadratic: State the full feature set.
Third quadratic: Explain why no real root is missed.
Higher-tier reciprocal: State domain and branch behaviour.
New transfer and error: Compare equivalent forms.
Evidence and graph limits: Retain exact mathematics and actual work.

Worked checks and assessment guidance

x²−6 x+8=(x −2)(x −4)=(x −3)²−1. Roots 2 and 4; turning point (3, −1); symmetry x=3; y-intercept (0, 8). Sample outputs 8, 3, 0, −1, 0, 3, 8 agree. At 2.5, the output−.75 is below the axis.
y=(x −1)² has one distinct repeated real root 1, turning point (1, 0) and symmetry x=1. Values 4, 1, 0, 1, 4 at inputs −1 through 3 agree. The y-intercept is (0, 1), not the vertex.
y=x²+1 has minimum 1 at 0, turning point (0, 1) and symmetry x=0. Since real squares are nonnegative, output can never be 0. Values 5, 2, 1, 2, 5 illustrate but do not replace that general argument.
For y=1/x, input 0 and output 0 are excluded. Negative inputs have negative outputs; positive inputs have positive outputs. Product xy=1 checks values such as−.5 with −2. Never add a point (0, 0) or connect branches across the forbidden input.
x²−4 x+3=(x −1)(x −3)=(x −2)²−1 gives roots 1 and 3, turning point (2, −1), symmetry x=2 and y-intercept (0, 3). For the source, squaring x −3 is minimised atx 3, not −3.
Keep three complete labelled sketches, checked tables, root-count arguments, reciprocal domain and unfamiliar transfer. Sampled points should be joined as a smooth quadratic, not a polygon. Neither an approximate drawing nor planned later practice proves an exact value or a later learning achievement.

Learning evidence

Keep three quadratic sketches with exact roots, turning points, symmetry and checked tables, plus the labelled reciprocal-domain extension, unfamiliar transfer and first-error repair.

OER and Wikipedia

Department for Education:GCSE Mathematics subject content and assessment objectives. Source area:GCSE Mathematics subject content > Scope of study > Algebra - Graphs items 11 to 16. Year10 allocation is editorial within the GCSE pathway;Higher-only extensions are explicitly labelled. Original examples,teaching and feedback;Commons images retain their actual attribution,and videos identify original publishers. Reviewed12November2026.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.

Illustration for the area of a rectangle.svg, Stephan Kulla, CC0. Unchanged original; caption and learning activities are original.