Quadratic graph features
Objective. Identify roots, intercepts, symmetry and turning points.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A quadratic graph is a parabola whose key features — roots, y-intercept, line of symmetry and turning point — can all be read from different forms of its equation. The roots are where the graph crosses the x-axis, found by setting y = 0 and factorising or using the formula; the y-intercept is the constant term when x = 0. The line of symmetry lies exactly halfway between the roots, and the turning point lies on this line, found most directly from completed square form a(x + p)² + q as (−p, q). A positive coefficient of x² gives a minimum turning point; a negative coefficient gives a maximum.
Worked example
A quadratic graph has roots at x = 2 and x = 8. Find the x-coordinate of its turning point using the graph's symmetry.
- 1The line of symmetry lies midway between the roots.
- 2Calculate the midpoint: (2 + 8) ÷ 2.
- 3(2 + 8) ÷ 2 = 5.
- 4The turning point has x-coordinate 5.
Answer: x = 5
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils find the roots correctly but then assume the turning point's x-coordinate is one of the roots rather than their midpoint. Always average the two roots to find the line of symmetry.
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