Quadratic Function y = ax²

Q: If y = 2x², what is y when x = 3?

18 — Square first, then multiply: y = 2 × 3² = 2 × 9 = 18.

Q: For y = ax² with a < 0, what is the y-value at the vertex?

The maximum (the vertex is the highest point) — When a < 0 every y-value is 0 or below, and at the vertex x = 0 you get y = 0. So the vertex is the largest value the function reaches.

Q: How does the graph of y = 3x² compare with y = x²?

y = 3x² is narrower (steeper) — The bigger the size of the coefficient, the faster y climbs, so the curve is pinched narrower and steeper.

Q: An object in free fall drops y = 5t² metres after t seconds. How far has it fallen after 3 seconds?

45 m — Substitute t = 3: y = 5 × 3² = 5 × 9 = 45 m.

Q: If y = −x², what is y when x = −4?

−16 — Square the −4 first, giving 16, and only then apply the minus sign in front: y = −(−4)² = −16.

Q: The curve y = ax² passes through the point (2, 12). What is a?

3 — Put x = 2 and y = 12 into the equation: 12 = a × 2² = 4a, so a = 3.

Q: For y = x², what is the rate of change as x goes from −2 to 4?

2 — Rate of change = (change in y) ÷ (change in x) = (4² − (−2)²) ÷ (4 − (−2)) = (16 − 4) ÷ 6 = 12 ÷ 6 = 2.

Q: Braking distance follows d = kv². At v = 60 km/h the distance is d = 36 m. What is d at v = 90 km/h?

81 m — Going from 60 to 90 km/h multiplies the speed by 1.5, so the distance is multiplied by 1.5² = 2.25. That gives 36 × 2.25 = 81 m.

Where this is used

Parabolas — curves of the form y = ax² — show up everywhere once you know to look. A car's braking distance grows with the square of its speed, which is exactly why highway speeds are so unforgiving. A camera's autofocus hunts along a curve shaped like a parabola to find the sharpest point. And a space telescope's mirror is a parabolic dish, because that shape gathers incoming light into a single spot.

Start gently

The graph of y = ax² is a parabola — the same shape as the path of a thrown ball. The larger a is, the narrower and sharper the curve.

Key points

The Basics of y = ax²

This is simply x² multiplied by a constant a. If a > 0 the parabola opens upward (a U shape); if a < 0 it opens downward. Either way the vertex — the turning point — sits at the origin (0, 0).

What the Graph Looks Like

Four things to remember: (1) the vertex is the origin (0, 0); (2) the curve is symmetric about the y-axis, the line x = 0; (3) the larger |a| is, the narrower and steeper the curve becomes; (4) since x² is never negative, a > 0 means y ≥ 0 always.

Rate of Change

A linear function has one fixed rate of change — its slope. A quadratic does not: the rate changes as x moves, which is what "nonlinear" means. Going from x = 1 to 3 and going from x = 3 to 5 cover the same width, but y climbs by different amounts.

Feel It Through Braking Distance

If it takes 9 m to stop from 30 km/h, then at 60 km/h (twice the speed) it takes 36 m — four times as far — and at 90 km/h (three times the speed) it takes 81 m, nine times as far. Distance grows with the square of speed, not with speed itself.

See it drawn

Change a and the parabola opens differently

Check yourself 10questions

Tap an option to check

Q1 If y = 2x², what is y when x = 3?

Q2 For y = ax² with a < 0, what is the y-value at the vertex?

Q3 How does the graph of y = 3x² compare with y = x²?

Q4 An object in free fall drops y = 5t² metres after t seconds. How far has it fallen after 3 seconds?

Q5 If y = −x², what is y when x = −4?

Q6 The curve y = ax² passes through the point (2, 12). What is a?

Q7 For y = x², what is the rate of change as x goes from −2 to 4?

Q8 Braking distance follows d = kv². At v = 60 km/h the distance is d = 36 m. What is d at v = 90 km/h?

Q9 How are the graphs of y = 2x² and y = −2x² related?

Q10 For y = ax² with a = 2, what is the largest value of y when x runs from −3 to 2?

Jobs that use this

Automotive Engineer (Safety Design) $100k

Optical Engineer $110k

Robotics & Drone Engineer $95k

Automotive Engineer (Safety Design)$100k

Uses y = ax² to calculate braking distances and crash energy. Because stopping distance grows with the square of speed, this maths feeds straight into vehicle safety standards.

Maths used: y = ax²CalculusPhysicsStatistics

Optical Engineer$110k

Designs the curvature of lenses and mirrors using the equation of a parabola. Day-to-day work means building telescopes, camera lenses and laser instruments.

Maths used: y = ax²TrigonometryWave equationsComplex numbers

Robotics & Drone Engineer$95k

Works out flight paths with the equations of projectile motion — where a package dropped from a drone will land, or how a self-driving robot should move.

Maths used: y = ax²VectorsControl engineeringMachine learning