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Grade 12 · Mathematics · Paper 1

Functions

Explore parabolas, hyperbolas, exponentials and their inverses — drag, type values and see the working.

Parabolas

Two equivalent forms:
y=a(x−p)2+q(turning point form)y = a(x-p)^2 + q \quad \text{(turning point form)}
y=ax2+bx+c(standard form)y = ax^2 + bx + c \quad \text{(standard form)}
Both describe the same parabola. Turning point form is more powerful for reading features directly off the equation.
Theorem 1📌 Learn for exam
Key features of y = a(x − p)² + q
  • Turning point (vertex): (p, q)(p,\, q)
  • Axis of symmetry: x=px = p
  • y-intercept: set x=0x = 0, giving y=ap2+qy = ap^2 + q
  • x-intercepts: set y=0y = 0 and solve a(x−p)2+q=0a(x-p)^2 + q = 0
  • Shape: if a>0a > 0 — minimum (smile ∪); if a<0a < 0 — maximum (frown ∩)
  • Range: if a>0a > 0: y≥qy \geq q; if a<0a < 0: y≤qy \leq q
  • Domain: always x∈Rx \in \mathbb{R}
Effect of each parameter:
  • ∣a∣|a| — controls vertical stretch. The larger ∣a∣|a|, the narrower the parabola.
  • a<0a < 0 — reflects the parabola about the x-axis (flips it upside down).
  • pp — shifts horizontally. Positive pp shifts right; negative pp shifts left. Note: the form is (x−p)(x - p), so the shift direction can be counter-intuitive.
  • qq — shifts vertically. Positive qq shifts up; negative qq shifts down.
▸ Parabola Explorer — drag a, p, q sliderssliders are interactive
Loading diagram…
Worked Example

Full analysis of a parabola

Analyse y = −2(x − 1)² + 8. Find the turning point, axis of symmetry, intercepts, range, and sketch.

Worked Example

Finding the equation from a graph

A parabola has turning point (2, −3) and passes through (0, 5). Find the equation.

Tip
In the exam, always identify aa, pp, and qq before answering any sub-question. Write them down explicitly — it prevents sign errors with pp.