Grade 9 · Grade 9: Algebra, Quadratics and Functions · Algebraic Structure, Quadratics and Growth
Quadratic Equations and Their Graphs
Solve quadratics four ways and read the vertex, intercepts and symmetry straight from the graph.
Learning objectives
- Solve quadratics by factorising, completing the square, the quadratic formula and graphing.
- Use the discriminant to count real solutions.
- Identify vertex, axis of symmetry, intercepts and end behaviour.
- Link algebraic form to graph features.
AERO Mathematics alignment
AERO.M9.AE.4
Solve quadratic equations in one variable by inspection, factorising, completing the square, using the quadratic formula and graphing, and recognise when there are no real solutions.
AERO.M9.AE.5
Graph quadratic functions, identifying the vertex, axis of symmetry, intercepts, maximum or minimum and end behaviour, and relate the algebraic form to the graph.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A ball is thrown upward and its height follows h = −5t² + 20t.
Think about it
When does the ball hit the ground, and when is it highest?
Hint: Factorise to find where the height is zero.
EXPLAIN
Solving and graphing quadratics
| Method | Best when | Example |
|---|---|---|
| Factorising | The expression factorises neatly | x² − 5x + 6 = 0 → x = 2 or 3 |
| Square roots | No linear term | x² = 49 → x = ±7 |
| Completing the square | You need the vertex | x² − 6x + 5 = (x − 3)² − 4 |
| Quadratic formula | Always works | x = (−b ± √(b² − 4ac)) / 2a |
The discriminant
b² − 4ac decides everything: positive gives two real solutions, zero gives one repeated solution, and negative gives none in the real numbers because the parabola never crosses the x-axis.
Reading the graph
Every quadratic graph is a parabola. The sign of a decides whether it opens up (minimum) or down (maximum). The axis of symmetry is x = −b/2a, and the vertex sits on it. For h = −5t² + 20t, factorising gives −5t(t − 4), so the ball lands at t = 4 s and reaches its maximum at t = 2 s, at a height of 20 m.
Common misconception
No real solution does not mean no answer. It means the parabola never crosses the x-axis, which in context can mean the ball never reaches that height.
INVESTIGATION
Try it yourself
Film or model a thrown object and record its height at several times. Fit a quadratic, then use your model to predict the maximum height and the landing time. Compare with what actually happened.
Watch
Solving Quadratic Equations Using the Quadratic Formula
The Organic Chemistry Tutor · 5:55
A worked walkthrough of the formula, including the discriminant.
Before you watch: What does it mean if the discriminant is negative?
- Solve 2x² + 3x − 5 = 0.
- Find the vertex of y = x² − 6x + 5.
Interactive simulation · PhET
Graphing Quadratics
Change a, b and c and record the effect on the parabola, vertex and roots.
While you explore
- What does a control?
- How does the graph look when the discriminant is negative?
Key vocabulary
- Vertex
- The highest or lowest point of a parabola.
- Discriminant
- The part of the quadratic formula that tells you how many solutions there are.
Practice questions
0/1 correct
Level 1 · Criterion A
How many real solutions does x² + 2x + 5 = 0 have?
MYP criterion tasks
Level 2 · Criterion C
Find the vertex of y = x² − 6x + 5 by completing the square, and state whether it is a maximum or minimum.
Reflect & track
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