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16.2 Plotting Quadratic Graphs

GCSE Maths Foundation · Edexcel · Chapter 16: Quadratic Equations and Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: tables of values, plotting smooth curves and recognising the shape of quadratic graphs. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Complete a table of values for a quadratic function
  • Plot the points and join them with a smooth curve
  • Recognise the ∪ and ∩ shapes of quadratic graphs
  • Find the line of symmetry, turning point and y-intercept of a quadratic graph

Key words

  • Quadratic function: A function such as y = x² − 2x − 3, where the highest power of x is 2.
  • Parabola: The ∪ or ∩ shaped curve of a quadratic graph.
  • Turning point: The lowest point (minimum) or highest point (maximum) of the curve.
  • Line of symmetry: The vertical line through the turning point that splits the curve into mirror halves.

Tables of values

Substitute each x value into the equation. Be careful with negative numbers: square first, and remember that a negative number squared is positive.

  • x²; −2 4; −1 1; 0 0; 1 1; 2 4; 3 9; 4 16
  • −2x; −2 +4; −1 +2; 0 0; 1 −2; 2 −4; 3 −6; 4 −8
  • −3; −2 −3; −1 −3; 0 −3; 1 −3; 2 −3; 3 −3; 4 −3
  • y; −2 5; −1 0; 0 −3; 1 −4; 2 −3; 3 0; 4 5

Worked example: y = x² − 2x − 3 when x = −2

(−2)² − 2 × (−2) − 3 = 4 + 4 − 3 = 5

Answer: y = 5, so plot the point (−2, 5)

Key fact: The y values of a quadratic are symmetrical. Here 5, 0, −3, −4, −3, 0, 5 mirror around x = 1. If one value breaks the pattern, check it.

Plotting the curve

Plot each point with a small cross. Join the points with one smooth curve drawn freehand, not with a ruler. The bottom of the curve is rounded, and it may dip slightly below the lowest plotted point if the turning point lies between two x values.

Worked example: reading features from the graph

Line of symmetry: halfway between the points where y = 0, x = −1 and x = 3, so x = 1.
Turning point (minimum): (1, −4). y-intercept: where x = 0, y = −3.

Answer: symmetry x = 1, minimum (1, −4), y-intercept −3

The shape of a quadratic graph

  • y = x² + … (positive x²); Shape ∪; Turning point a minimum
  • y = −x² + … (negative x²); Shape ∩; Turning point a maximum

Worked example: y = 6 + x − x²

The x² term is negative, so the graph is ∩ shaped.
x = −2: 6 − 2 − 4 = 0; x = 0: 6; x = 3: 6 + 3 − 9 = 0.

Answer: ∩ shape crossing the x-axis at −2 and 3, y-intercept 6

Common mistakes

  • Working out (−3)² as −9. A negative number squared is positive: (−3)² = 9.
  • Joining the points with straight lines. A quadratic graph is a smooth curve.
  • Drawing a flat bottom between two equal lowest points instead of a rounded curve.
  • Plotting a point that breaks the symmetry without checking the calculation.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out the value of y = x² - 3x + 1 when x = -2.

Answer: 11

Working: 4 + 6 + 1 = 11.

Question 2. What shape is the graph of y = 7 - x²? (Choose from: ∩ shape with a maximum point, ∪ shape with a minimum point, A straight line, A ∪ shape that never crosses the x-axis.)

Answer: ∩ shape with a maximum point

Working: The x² term is negative, so the graph is ∩ shaped with a maximum at (0, 7).

Question 3. A quadratic graph crosses the x-axis at x = -1 and x = 5. Its line of symmetry is x = k. Find k.

Answer: 2

Working: (-1 + 5) ÷ 2 = 2, so the line of symmetry is x = 2.

Related lessons

Continue in the full course

This lesson is section 16.2 (EGCMF1602) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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