Quadratic Graphs and Features
Move between quadratic equations and sketches, identifying intercepts, turning point, symmetry and orientation.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
Factorised form makes x-intercepts visible, while completed-square form makes turning point and axis of symmetry visible.
Exam cue: Select the equation form that reveals the requested feature.
Concept 2
The sign of the leading coefficient determines whether the parabola has a minimum or maximum turning point.
Exam cue: Find the y-intercept by setting x = 0.
Concept 3
A valid sketch shows shape and key features without needing an exact plotting scale.
Exam cue: Label roots, turning point and axis consistently on the sketch.
Risk pitfalls and guardrails
Reading the turning point signs directly from bracket notation without reversing the horizontal sign.
Guardrail: Check signs, coefficients and both sides of every algebraic transformation.
Drawing a graph that contradicts the leading coefficient.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Omitting the y-intercept or axis when explicitly requested.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Factorised form
The factors reveal the x-intercepts.
Completed-square form
y = k(x + p)² + q has turning point (−p, q).
Axis of symmetry
x = −p for y = k(x + p)² + q.
Leading coefficient
Positive opens upward; negative opens downward.
y-intercept
Set x = 0.
Sketch check
Shape, roots, y-intercept, turning point and symmetry agree.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find the x-intercepts of y = (x − 2)(x + 5).
Find the y-intercept of y = x² − 3x − 4.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
What is Pass Harbor?
Completely free exam prep for 247 UK exams.
- Practice questions
- Flashcards
- Study guides
- Mock exams
- No registration
- No paywall
- Start instantly
“No more expensive exam prep. Quality study tools should be accessible to everyone.”