IGCSE Maths · Tool
Graphing calculator
How do you find the roots and turning point of a quadratic graph?
The roots of y = ax² + bx + c are where the curve crosses the x-axis, so they solve ax² + bx + c = 0. The turning point lies on the line of symmetry, x = −b ÷ 2a, halfway between the roots; put that x into the equation to find its y. A curve that never reaches the x-axis has no real roots.
y = x² − 2x − 3
- Discriminant
- b² − 4ac = (−2)² − 4 × 1 × (−3) = 16
- Roots, where y = 0
- x = −1 or x = 3
- The discriminant is positive, so the curve crosses the x-axis twice.
- Turning point
- (1, −4)
- A minimum, because a is positive.
- y-intercept
- (0, −3)
Free to use here, with no account.
What to take away
- The roots of a quadratic are where its graph crosses the x-axis.
- The turning point is on the line of symmetry, x = −b ÷ 2a, halfway between the roots.
- When a is positive the curve has a minimum point; when a is negative, a maximum.
Common mistakes
- Reading the y-intercept, c, as a root.
- Expecting every quadratic to have two roots: one that never meets the x-axis has none.
Questions
How do you find the roots and turning point of a quadratic graph?
The roots of y = ax² + bx + c are where the curve crosses the x-axis, so they solve ax² + bx + c = 0. The turning point lies on the line of symmetry, x = −b ÷ 2a, halfway between the roots; put that x into the equation to find its y. A curve that never reaches the x-axis has no real roots.
Where does a quadratic graph cross the y-axis?
At y = c, the constant term, because every other term of y = ax² + bx + c is 0 when x = 0.
Is the graphing calculator free?
Yes. It runs here in your browser with no account, and every ConceptOrbit course tool is free.
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