16.3 Using Quadratic Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: reading solutions, turning points and values from quadratic graphs, including real-life 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

• Use a graph to solve a quadratic equation such as x² − x − 6 = 0
• Solve equations such as x² − x − 6 = 4 by drawing a horizontal line
• Read the turning point and estimate values from a quadratic graph
• Interpret quadratic graphs in real-life contexts

Key words

• Root: A solution of an equation. On a graph of y = …, the roots of … = 0 are where it crosses the x-axis.
• Turning point: The minimum or maximum point of a quadratic graph.
• Estimate: A value read from a graph. It is approximate, usually to the nearest small square.
• Intersection: A point where two graphs cross.

Solving quadratic equations with a graph

On the graph of y = x² − x − 6, the points where y = 0 lie on the x-axis. Their x values are the solutions (roots) of x² − x − 6 = 0.

Worked example: x² − x − 6 = 0

Read where the curve crosses the x-axis: at x = −2 and x = 3.
Check: (−2)² − (−2) − 6 = 4 + 2 − 6 = 0 ✓ and 9 − 3 − 6 = 0 ✓

Answer: x = −2 or x = 3

Solving … = k with a horizontal line

To solve x² − x − 6 = 4, draw the line y = 4. Read down from the points where it meets the curve.

Worked example: x² − x − 6 = 4

The line y = 4 crosses the curve at x ≈ −2.7 and x ≈ 3.7.
Check one: 3.7² − 3.7 − 6 = 13.69 − 9.7 = 3.99, which is close to 4.

Answer: x ≈ −2.7 or x ≈ 3.7

Key fact: A horizontal line can meet a ∪ curve twice (two solutions), once at the turning point (one solution) or not at all below the minimum (no solutions).

Turning points and reading values

The turning point lies on the line of symmetry, halfway between the roots. For y = x² − x − 6 the roots are −2 and 3, so the line of symmetry is x = (−2 + 3) ÷ 2 = 0.5.

Worked example: turning point

x = 0.5: y = 0.25 − 0.5 − 6 = −6.25

Answer: minimum point (0.5, −6.25)

Worked example: number of solutions

How many solutions does x² − x − 6 = −8 have?
The lowest value of y is −6.25. The line y = −8 is below the curve and never meets it.

Answer: no solutions

Real-life quadratic graphs

Heights of thrown objects, areas and profits often give quadratic graphs. Read the question carefully: the turning point is often the greatest height or the biggest profit.

Worked example: a firework

A firework's height is h = 40t − 5t² metres after t seconds.
It lands when h = 0: 40t − 5t² = 5t(8 − t) = 0, so t = 0 or t = 8.
Greatest height at the line of symmetry t = 4: h = 160 − 80 = 80.

Answer: greatest height 80 m after 4 s; it lands after 8 s

Common mistakes

• Reading the y-intercept instead of the x-intercepts when solving … = 0.
• Giving only one solution when the line crosses the curve twice.
• Reading the wrong scale: check how much one small square is worth on each axis.
• Forgetting that answers read from a graph are estimates. Check by substituting.

Revision questions

Try these, then check your answers below each one.

Question 1. What is the y-coordinate of the turning point of y = x² - 4x?

Answer: -4

Working: x = 2: y = 4 - 8 = -4.

Question 2. How many solutions does x² - 4x = -6 have? (Choose from: 0, 1, 2, 3.)

Answer: 0

Working: The line y = -6 is below the minimum value -4, so it never meets the curve: no solutions.

Question 3. Kim wants to solve x² + x - 6 = 4 using the graph of y = x² + x - 6. She reads where the curve crosses the x-axis. Is she correct?

Answer: No

Working: She should draw the line y = 4 and read where it meets the curve. The x-axis gives solutions of x² + x - 6 = 0.

Related lessons

• Previous lesson: 16.2 Plotting Quadratic Graphs
• Next lesson: 16.4 Factorising Quadratic Expressions
• In this chapter: 16.1 Expanding Double Brackets
• In this chapter: 16.5 Solving Quadratic Equations Algebraically

Continue in the full course

This lesson is section 16.3 (EGCMF1603) 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.