Sketching & Transforming Graphs

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AS Maths – Pure – Cubic Graphs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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4A Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (Up to 6:40)

Bicen Maths Live Lesson

Graphs & Transformations 1 • Shapes of Polynomials • P1 Ex4A • 🤖

Graphs & Transformations 2 • Sketching Cubics • P1 Ex4A • 🤖

Graphs & Transformations 3 • Finding the Equation from a Graph • P1 Ex4A • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 1-12✦

Extra Questions

• Dr Frost Questions (Short but nice final question extending into higher order polynomials)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 494a: Identify the intercepts of a cubic with the coordinate axes, where the equation is factorised.
• 494b: Sketch a cubic graph where the function is in factorised form (including cubics with a stationary point of inflection).
• 494c: Sketch a cubic graph where the function requires factorising.
• 494d: Identify the factorised equation of a sketched cubic graph.
• 494e: Identify the expanded equation of a sketched cubic graph.
• 427c: Identify a particular cubic graph amongst other cubic graphs.
• 494: Exam Practice: Graphs of polynomials of order 3 or more, particularly cubic and quartic functions
• 427: Exam Practice: Recognising the shape of more complex forms of quadratic, cubic, reciprocal and exponential graphs based on their equations

Other Videos

Sketching Graphs

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Underground Mathematics Can you find… cubic edition✦ 
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q20
• Desmos
  • Polynomial Equation Challenges (write cubics and quartics to match given roots and points)
• SRWhitehouse
  • Matching activity for cubic graphs✦

Worksheets

• Jethwa Maths Starter Activities (p1-2)
• Susan Wall Cubic graphs✦
• hannahlees Cubics – fill in the gaps (TES Account Required)
• MEI Equations of cubic curves
• Mathshelper (Jon Stone)
  • Polynomial Factorising & Sketching
• Madas Maths
  • Sketching cubics booklet✦
  • Cubics exam questions✦
• StudyWell
  • Cubics worksheet✦
  • Cubics exam questions with solutions✦
• MadAsMaths (T. Madas)
  • Cubic Sketching✦ Q1–19
  • Cubics Exam Questions✦ Q1–3, 5–6, 8–10, 12–16, 21, 24 (difficulty 2–4 of 5)
  • Curve Sketching Q2, 5, 7, 18, 20, 46 (difficulty 2–4 of 5)

AS Maths – Pure – Quartic Graphs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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4B Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (From 6:40 to 10:25)

Bicen Maths Live Lesson

Graphs & Transformations 4 • Sketching Quartics • P1 Ex4B • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 13-17✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 494f: Sketch the graph of a quartic function where the function is in factorised form (including quartics with a stationary point of inflection).
• 494g: Sketch the graph of a quintic function where the function is in factorised form.
• 494: Exam Practice: Graphs of polynomials of order 3 or more, particularly cubic and quartic functions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Polynomial Graphs

Worksheets

• Jethwa Maths Starter Activities (p5-6)
• kutasoftware Worksheets
  • Graphs, Real Zeros, and End Behavior (some knowledge of factor theorem would be beneficial – this worksheet has also been added to that section)
• MadAsMaths (T. Madas)
  • Polynomials Exam Questions (Intro) Q38 (difficulty 3 of 5)
  • Curve Sketching Exam Questions Q10 (difficulty 3 of 5)
  • Curve Sketching Q4, 19, 61 (difficulty 3–4 of 5)

AS Maths – Pure – Reciprocal Graphs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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4C Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (From 10:25)

Bicen Maths Live Lesson

Graphs & Transformations 5 • Sketching Reciprocal Graphs • P1 Ex4C • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 18-21✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions (Very Short)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 493a: Identify the x and y intercepts of a more general reciprocal function.
• 493b: Identify the equation of the vertical asymptote of a more general reciprocal function.
• 493c: Identify the equation of the horizontal asymptote of a more general reciprocal function.
• 493d: Sketch reciprocal functions given in the form y = a + b/cx ± d
• 493e: Sketch graphs given in the form y = a/x² (not yet active)
• 493f: Sketch reciprocal functions given in the form y = ax ± b/cx ± d
• 493g: Determine coefficients in a more general reciprocal function that lead to a given x or y intercept.
• 493h: Determine coefficients in a more general reciprocal function that lead to a given vertical or horizontal asymptote.
• 427d: Identify a particular reciprocal graph amongst other reciprocal graphs.
• 493: Exam Practice: Graphs of more general reciprocal functions
• 427: Exam Practice: Recognising the shape of more complex forms of quadratic, cubic, reciprocal and exponential graphs based on their equations

Practice Questions

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• MathsPad Functions Stack✦
• Math = Love (Sarah Carter)
  • Finding Asymptotes Foldable
• Desmos
  • Marbleslides: Rationals (transform reciprocal graphs so the marbles pass through the stars)
• martahidegkuti
  • Graphing Reciprocal Functions✦

Worksheets

• kutasoftware Worksheets
  • Graphing Simple Rational Functions (Some Knowledge of Transformations Required)
• Mathshelper (Jon Stone)
  • Reciprocal Sketching
• Dr Austin Maths
  • Sketching Reciprocal Graphs Practice Grid (Answers)✦

AS Maths – Pure – Points of Intersection

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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4D Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video

Bicen Maths Live Lesson

Graphs & Transformations 6 • Points of Intersections on Graphs pt.1 • P1 Ex4D • 🤖

Graphs & Transformations 7 • Points of Intersections of Graphs pt.2 • P1 Ex4D • 🤖

Graphs & Transformations 8 • Points of Intersections, Unknown Constants • P1 Ex4D • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 22-25✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 421a: Determine the number of solutions to f(x)=g(x) given graphs of y=f(x) and y=g(x)
• 421b: Determine the number of solutions to an equation that can be rearranged to f(x)=g(x) given graphs of y=f(x) and y=g(x)
• 421c: Determine the range of values for k in f(x)=k given the graph of y=f(x) and the number of solutions.
• 421i: Draw a straight line to determine approximate solutions to a cubic equation.
• 422g: Determine the number of solutions to an equation involving a reciprocal term and a straight line or quadratic curve.
• 421: Exam Practice: Using graphs to count solutions and find approximate solutions to equations
• 422: Exam Practice: Plotting reciprocal graphs

Practice Questions

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• PMT OCR MEI C1 Factorising & Sketching Polynomials 1 PPQs & MS
• PMT OCR MEI C1 Factorising & Sketching Polynomials 2 PPQs & MS
• PMT OCR MEI C1 Factorising & Sketching Polynomials 3 PPQs & MS
• PMT OCR MEI C1 Factorising & Sketching Polynomials 4 PPQs & MS
• MadAsMaths (T. Madas)
  • Cubics Exam Questions✦ Q7, 17–18, 25 (difficulty 3–4 of 5)
  • Equations Exam Questions I Q42 (difficulty 4 of 5)
  • Polynomials Exam Questions (Intro) Q52 (difficulty 3 of 5)
  • Reciprocal Functions Q12, 22 (difficulty 3–4 of 5)

AS Maths – Pure – Translating Graphs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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4E Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (4E, F & G)

Bicen Maths Live Lesson

Graphs & Transformations 9 • Translations • P1 Ex4E • 🤖

Graphs & Transformations 10 • Translations, problem solving • P1 Ex4E • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 26-32✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 448a: Sketch the graph y = f(x + a) given the graph of y = f(x) (not yet active)
• 448b: Sketch the graph y = f(x) + a given the graph of y = f(x) (not yet active)
• 448c: Understand the effect on a point under the transformation y = f(x + a)
• 448d: Understand the effect on a point under the transformation y = f(x) + a
• 448e: Apply transformations y=f(x+a) and y=f(x)+a in a table. (not yet active)
• 448g: Determine the equation in f(x) after a translation of a given graph to f(x)+a and f(x+a)
• 448h: Describe the translation of a function given the equations before and after. (not yet active)
• 448i: Determine the new equation of a function after a translation by beginpmatrixa0endpmatrix or beginpmatrix0aendpmatrix
• 451a: Sketch the graph y = f(x + a) + b given the graph of y = f(x)
• 451h: Determine the equation in f(x) notation after a translation of a given graph to y = f(x + a) + b (not yet active)
• 451k: Describe the translation of a function y=x² to y=(x+a)²+b (not yet active)
• 451l: Determine the new equation of a function after a translation by a given vector beginpmatrixabendpmatrix (not yet active)
• 448: Exam Practice: Graph transformations of y = f(x+a) and y = f(x) + a
• 451: Exam Practice: Combinations of graph transformations

Other Videos

Transforming Graphs

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

AS Maths – Pure – Stretching Graphs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

4F Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (4E, F & G)

Bicen Maths Live Lesson

Graphs & Transformations 11 • Stretches • P1 Ex4F • 🤖

Graphs & Transformations 12 • Reflections or 'Flippings' • P1 Ex4F • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 26-32✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 449a: Sketch a graph of y = -f(x) given the graph of y = f(x)
• 449b: Sketch a graph of y = f(-x) given the graph of y = f(x)
• 449c: Understand the effect on a point under the transformation y = -f(x)
• 449d: Understand the effect on a point under the transformation y = f(-x)
• 449f: Determine the equation in f(x) notation after a reflection of a given graph in one axis only.
• 449g: Determine the new equation of a function after a reflection in one axis only.
• 449h: Describe a reflection given the equations of the original and transformed functions. (not yet active)
• 450a: Sketch a graph of y = af(x) given the graph of y = f(x)
• 450b: Sketch a graph of y = f(ax) given the graph of y = f(x)
• 450c: Understand the effect on a point under the transformation y = af(x)
• 450d: Understand the effect on a point under the transformation y = f(ax)
• 450e: Apply transformations y=f(ax) and y=af(x) in a table. (not yet active)
• 450g: Determine the new equation in f(x) notation after the transformation of a given graph to y=af(x) or y=f(ax)
• 450h: Determine the new equation of a function after a scaling transformation y=f(ax) or y=af(x) (not yet active)
• 450i: Determine a vertical scaling on an exponential graph equivalent to a horizontal translation, or vice versa.
• 449: Exam Practice: Graph transformations of y = f(-x) and y = -f(x)
• 450: Exam Practice: Graph transformations of y = f(ax) and y = af(x) (dilations)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

AS Maths – Pure – Transforming Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

4G Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (4E, F & G)

Bicen Maths Live Lesson

Graphs & Transformations 13 • Summary of Transformations, memory tips • P1 Ex4G • 🤖

Graphs & Transformations 14 • Effects on Coordinates • P1 Ex4G • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 33-35✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 451b: Sketch the graph y = af(bx) given the graph of y = f(x) (not yet active)
• 451c: Sketch the graph af(x)+b, af(x+b) and f(ax)+b given the graph of y = f(x)
• 451i: Determine the equation in f(x) notation after the transformation of a given graph to y = af(bx) (not yet active)
• 451j: Determine the equation in f(x) notation after the transformation of a given graph to y = af(x)+b, y = af(x+b) or y = f(ax)+b (not yet active)
• 451m: Determine the new equation of a function after transformations given in the form y = af(bx) (not yet active)
• 451n: Determine the new equation of a function after a combination of transformations, y = af(x)+b, y = af(x+b) and y = f(ax)+b (not yet active)
• 452a: Sketch the graph of y=x^1/2 or y=√x (not yet active)
• 452b: Sketch the graph of y=ax^1/2 or y=a√x (not yet active)
• 452c: Sketch the graph of y=ax^-1/2 or y=fraca√x (not yet active)
• 452d: Sketch a graph of y = a√x+b (not yet active)
• 452e: Sketch a graph of y = asqrt[3]x+b (not yet active)
• 451: Exam Practice: Combinations of graph transformations
• 452: Exam Practice: Graphs of root functions

Practice Questions

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Transformations of Functions

Activities

• WW Transformations of Graphs Booklet Word Doc & PDF
• The Chalk Face Graph Transformations
• The Chalk Face Trig Transformations
• Median Don Steward Graph Transforms✦
• Underground Mathematics Translating or Not?✦, Order! Order!✦
• MEI Understanding Transformations
• Ben Sparkes Geogebra Interactive Activity

Worksheets

• PMT OCR C1 Transformation of Graphs PPQs & Solutions (Old Spec)
• PMT OCR MEI C1 Curve Sketching – Transformations 1 PPQs & MS (Old Spec)
• PMT OCR MEI C1 Curve Sketching – Transformations 2 PPQs & MS (Old Spec)
• StudyWell
• kutasoftware Worksheets
  • Graphing Simple Rational Functions (Some Knowledge of Transformations Required)
• MadAsMaths (T. Madas)
  • Curve Sketching Exam Questions Q5 (difficulty 3 of 5)
  • Curve Sketching Q13, 27, 31, 32, 34, 39–41 (difficulty 3–4 of 5)

Activities

• Durham County Council Graphs Mystery✦
• NRICH (student problems)
  • Coordinate Geometry & Graphs
• Underground Mathematics
  • curve match✦
• BossMaths
  • Sketching graphs applet✦
• Andy Lutwyche
  • Erica’s Errors Functions

External Whole Topic Past Paper Style Questions

• Jethwa Maths Worksheet (p13-18, *Q6 is with ln), Topic Test (p12-16)
• Dr Frost Topic Test✦ (Teacher Log in Required)
• Mathsaurus Mixed Exam Board PPQs
• PMT
  • Summary Graphs & Transformations PPQs
  • Edexcel Set 1 Functions – Transformation & Graphs  PPQs (Old Spec)
  • Edexcel Set 2 Curve Sketching PPQs (Old Spec)
  • OCR MEI Inequalities PPQs & Solutions (Old Spec)
  • Summary Cheat Sheet

Worksheets

• Mathshelper (Jon Stone)
  • Graphical Transformations
  • Polynomial Equation Finding
  • Polynomial Sketching
• MathedUp
  • Graphs transformations: Questions
• Maths Genie
  • Graphs and Transformations: Questions · Solutions · Lesson · Topic page
• Naiker Maths
  • Graphs & Transformations: Exam Qs 2005–11 · Mark scheme · Exam Qs 2012–17 · Mark scheme
• Physics & Maths Tutor
  • Transformations and Graphs: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• MME
  • C1 Curve Sketching: Questions | Answers  •  C2 Curve Sketching: Questions | Answers  •  Proportional Relationships & their Graphs: Questions | Answers
• MadAsMaths (T. Madas)
  • Cubics Exam Questions✦ Q19, 26, 29 (difficulty 4–5 of 5)
• Dr Austin Maths
  • Cubic Practice Grid (Word | PDF✦ | Answers✦)
  • Quartic Practice Grid (Word | PDF✦ | Answers✦)
  • Finding Equations of Cubics and Quartics from their Practice Grid (Word | PDF✦ | Answers✦)
  • Reciprocal Practice Grid (Word | PDF✦ | Answers)
  • Translations Fill in the Blanks (Word | PDF✦ | Answers)
  • Stretches Fill in the Blanks (Word | PDF✦ | Answers✦)