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20.2 Non-linear Graphs

GCSE Maths Foundation · Edexcel · Chapter 20: More Algebra

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: quadratic graph features and curved graphs in real life. 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

  • Recognise linear, quadratic, cubic and reciprocal graphs from their equations
  • Find the roots, y-intercept, turning point and line of symmetry of a quadratic graph
  • Use a quadratic graph to solve equations
  • Interpret curved graphs in real-life contexts, such as filling containers and moving objects

Key words

  • Non-linear: Any graph that is not a straight line.
  • Roots: The x values where a graph crosses the x-axis. They solve the equation y = 0.
  • Turning point: The point where a curve changes direction: the lowest point of a U or the highest point of a ∩.
  • Line of symmetry: For a quadratic graph, the vertical line through the turning point.

Which type of graph?

You can tell the shape of a graph from its equation, without plotting it.

  • x but no higher power; Type linear; Shape straight line
  • x² (and no higher power); Type quadratic; Shape U shape, or ∩ if the x² term is negative
  • x³; Type cubic; Shape S-shaped curve
  • x on the bottom: k/x; Type reciprocal; Shape two separate curves

Worked example: classify

y = 7 − 2x: linear. y = 3x² + 1: quadratic (U). y = 10/x: reciprocal. y = 5 − x²: quadratic (∩).

Answer: linear, quadratic, reciprocal, quadratic

Features of a quadratic graph

A quadratic graph is symmetrical. The line of symmetry passes through the turning point, halfway between the two roots.

Worked example: reading the features

Roots: the curve crosses the x-axis at x = −1 and x = 3.
y-intercept: put x = 0, y = −3, so (0, −3).
Line of symmetry: halfway between −1 and 3, so x = 1.
Turning point: x = 1 gives y = 1 − 2 − 3 = −4, so (1, −4).

Answer: roots −1 and 3; (0, −3); x = 1; minimum point (1, −4)

Solving equations with a quadratic graph

To solve x² − 2x − 3 = k, draw the horizontal line y = k and read the x values where it meets the curve.

Key fact: If the line misses the curve, there are no solutions. If it just touches the turning point, there is one solution.

Curved graphs in real life

Water poured at a steady rate rises steadily in a straight-sided container. In a container that gets wider, the depth rises more and more slowly. In one that gets narrower, it rises faster and faster.

Worked example: match the containers

A has straight sides: steady rise, graph 1 (straight line).
B gets wider: the rise slows down, graph 2 (getting less steep).
C gets narrower: the rise speeds up, graph 3 (getting steeper).

Answer: A–1, B–2, C–3

Worked example: a thrown ball

The highest point of the curve is at t = 2, h = 20.
The ball is back on the ground when h = 0 again, at t = 4.

Answer: maximum 20 m after 2 s; lands after 4 s

Common mistakes

  • Giving the turning point as a single number. It is a point, so give both coordinates.
  • Giving only one root when a U-shaped graph crosses the x-axis twice.
  • Mixing up "getting steeper" and "getting less steep" for filling containers. Wide parts fill slowly.
  • Thinking a graph is linear just because it has no x² written first, e.g. y = 4 − x² is still quadratic.

Revision questions

Try these, then check your answers below each one.

Question 1. What type of graph is y = 5 - x²? (Choose from: Quadratic, ∩-shaped, Quadratic, U-shaped, Linear, Cubic.)

Answer: Quadratic, ∩-shaped

Working: The highest power is 2, so it is quadratic. The x² term is negative, so it is ∩-shaped.

Question 2. Where does the graph of y = x² + 3x - 10 cross the y-axis? Give the y value.

Answer: -10

Working: 0 + 0 - 10 = -10

Question 3. Is Ali correct? Ali says, "The graph of y = x² - 16 crosses the x-axis only at x = 4."

Answer: No

Working: (-4)² - 16 = 0 as well, so it crosses at x = -4 and x = 4.

Related lessons

Continue in the full course

This lesson is section 20.2 (EGCMF2002) 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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