Roots
Where the graph crosses the x-axis. At these points, y = 0.
Learning Objectives
- recognise a quadratic function;
- complete a table of values for y = ax² + bx + c;
- draw a smooth quadratic graph;
- identify roots, the y-intercept and turning point;
- solve simple equations graphically.
Introduction
This lesson develops Quadratic Graphs from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🤔 Think About This...
A ball thrown into the air rises, slows down, reaches a highest point and then falls.
Would its height against time form a straight line?
No. The rate of change is not constant, so the graph curves.
∪
Change the Quadratic
Adjust the coefficient and constant to see how the parabola responds.
What Is a Quadratic Graph?
A quadratic function contains an x² term. Its graph is called a parabola.
y = x² − 4
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
A quadratic graph is smooth and curved. Do not join its points using
separate straight line segments.
Important Features
y-intercept
Where x = 0.
Turning point
The lowest or highest point of the curve.
Axis of symmetry
A vertical line dividing the parabola into matching halves.
Worked Example: Reading Roots
The graph of y = x² − 4 crosses the x-axis at x = −2 and x = 2.
1
At an x-intercept, y = 0.
2
Therefore x² − 4 = 0.
3
The graph shows the solutions x = −2 and x = 2.
Solving Equations Graphically
To solve x² − 4 = 3, find where the graph y = x² − 4 has height y = 3.
- Draw or imagine the horizontal line y = 3.
- Read the x-coordinates where it meets the parabola.
- Those x-values solve the equation.
Graphical solutions are usually approximate. Read values to a sensible
degree of accuracy.
✅ Quick Check
What shape is a quadratic graph?
What is true about y at a root?
Find y when x = −2 for y = x² + x − 1.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Complete a table for y = x² − 1 from x = −3 to x = 3.
- Plot the points and draw a smooth curve.
- Read the roots from your graph.
- Draw y = x² − 2x − 3 for −2 ≤ x ≤ 4.
- Estimate the roots and turning point.
- Use the graph to solve x² − 2x − 3 = 2.
- Explain why the graph of y = −x² + 4 opens downward.
✅ Check Your Work
Attempt the questions before opening the solutions.
Show answers and working
Foundation
- For x=−3,−2,−1,0,1,2,3, y=8,3,0,−1,0,3,8.
- The points form a smooth U-shaped curve.
- The roots are x=−1 and x=1.
Developing
- For x=−2,−1,0,1,2,3,4, y=5,0,−3,−4,−3,0,5.
- Roots: x=−1 and x=3. Turning point: (1,−4).
- Solve x²−2x−3=2, giving x²−2x−5=0. Graphically x≈−1.45 and x≈3.45.
Challenge
- The coefficient of x² is negative, so y decreases away from its maximum and the parabola opens downward.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- A quadratic function contains x².
- Its graph is a smooth parabola.
- Roots occur where y = 0.
- Graphs can solve quadratic equations approximately.