Quadratic graphs: roots, intercepts, turning points and completing the square
A quadratic y = ax² + bx + c is a parabola. Its key features — roots, y-intercept, turning point, and line of symmetry — are all directly readable from suitable algebraic forms.
y-intercept
Set x = 0: y = c. Trivial.
Roots (x-intercepts)
Set y = 0: ax² + bx + c = 0. Solve by factorising, completing the square, or the quadratic formula. The roots are where the curve crosses the x-axis.
A parabola has 0, 1 or 2 real roots depending on the discriminant b² − 4ac.
Turning point — by completing the square [H]
Rewrite the quadratic as y = a(x − h)² + k. The turning point is at (h, k):
- if
a > 0, parabola opens up and(h, k)is the minimum. - if
a < 0, parabola opens down and(h, k)is the maximum.
The line of symmetry is x = h.
Completing the square — the technique
For x² + bx + c:
- Take half of b, square it:
(b/2)². - Write
x² + bx + c = (x + b/2)² − (b/2)² + c.
Example: x² + 6x + 11. Half of 6 is 3, squared is 9.
= (x + 3)² − 9 + 11 = (x + 3)² + 2. Turning point: (-3, 2).
For ax² + bx + c with a ≠ 1, factor a out of the first two terms first:
2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3. Turning point (-2, -3).
Sketching strategy
- Plot the y-intercept (0, c).
- Solve for roots; plot any (x, 0) crossings.
- Find turning point via completion of the square (or by symmetry of roots: midpoint of roots is the x-coordinate of the vertex).
- Decide direction of opening from the sign of a.
- Sketch a smooth U or ∩ through the points.
✦Worked example
Sketch y = x² − 4x − 5.
- y-intercept: c = -5.
- Roots: factor as
(x − 5)(x + 1) = 0→ x = 5 or x = -1. - Vertex: midpoint of roots = (5 + (-1))/2 = 2; substitute → y = 4 − 8 − 5 = -9. So vertex (2, -9).
- a = 1 > 0, opens upwards.
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting the sign in completing the square.
x² − 6x = (x − 3)² − 9, NOT(x − 3)² + 9. Always subtract(b/2)². - Botching the factoring of a. When
a ≠ 1, only factor a out of the x² and x terms — leave the constant alone until the end. - Reading vertex sign wrong.
(x − 3)²puts the vertex at x = +3, not -3. - Confusing min/max with roots. Roots are where y = 0; vertex is where the parabola turns.
- Forgetting symmetry. The line of symmetry passes through the vertex and is parallel to the y-axis.
➜Try this— Quick check
y = x² + 8x + 7. Complete the square: (x + 4)² − 16 + 7 = (x + 4)² − 9. Turning point (-4, -9), minimum (a > 0).
AI-generated · claude-opus-4-7 · v3-deep-algebra