Watch out
Sketching a straight line instead of a curve.
Back to National 5 Mathematics
SQA National 5 Mathematics
Quadratic graphs
Recognising parabolas, roots, intercepts and turning points.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Recognise the shape of a parabola.
- Know that roots are x-intercepts.
- Know the y-intercept is where x = 0
Explanation
A quadratic graph has equation y = ax² + bx + c and forms a parabola.
The graph opens upwards when a is positive and downwards when a is negative.
Roots, the y-intercept and the turning point are the key features used for sketches.
Visual support
Parabola features
Key formulae and rules
- y = ax² + bx + c
- roots occur when y = 0
- y-intercept occurs when x = 0
Check
Substitute each solution back into the equation. A quick check catches most sign errors.
Exam tip
A good sketch needs labelled key points; it does not need graph-paper accuracy unless asked.
Calculator tip
Enter the numerator in brackets, especially when using a negative b or the ± answers separately.
Worked examples
Worked example 1
Find intercepts
Find the intercepts of y = x² − 5x + 6.
- Factorise: y = (x − 2)(x − 3)
- Roots are x = 2 and x = 3
- When x = 0, y = 6
Answer: x-intercepts are (2, 0) and (3, 0). y-intercept is (0, 6).
Worked example 2
Use symmetry
Find the axis of symmetry for y = x² − 6x + 8.
- The roots are x = 2 and x = 4.
- The axis is halfway between roots.
- (2 + 4) / 2 = 3
So: Axis of symmetry is x = 3
Worked example 3
Find the turning point
For y = x² − 6x + 8, find the turning point.
- Axis of symmetry is x = 3
- Substitute x = 3
- y = 9 − 18 + 8 = −1
Answer: Turning point is (3, −1).
Watch out
- Sketching a straight line instead of a curve.
- Forgetting that roots are where y = 0
- Using the y-intercept as the turning point.