Why does anyone care about parabolas?
A parabola is what you get whenever one quantity depends on the square of another. That turns out to be very often: the path of anything thrown or fired, because gravity pulls at a constant rate while horizontal speed stays the same; the area of a rectangle as you change one side; the revenue when raising the price loses you customers; the shape of a satellite dish or a headlight reflector, because a parabola is the only curve that focuses parallel rays to a single point.
All of those questions have the same form: where is the turning point? The highest the ball gets, the price that earns most, the dimensions that enclose most area. So the reason the course spends so long moving between forms is that the turning point is invisible in one of them and written down in another.
The second question is where does it cross zero? When does the ball land, at what price do you break even. That one is easy from factored form and hard from the others. Neither form is better; they answer different questions.
Where it turns up
- Throwing anything — height against time is a parabola, so the peak is a vertex and the landing is a root
- Pricing — revenue is price times quantity, and quantity falls as price rises — a product of two linear things, which is quadratic
- Headlights and dishes — the parabola's focus property is why the reflector behind a bulb is that shape
- Bridges and arches — a suspension cable carrying an even load hangs in a parabola
- 1. The words, first
- 2. What each form is good for
- 3. Moving between forms
- 4. Solving a real question end to end
- 5. Sanity checks worth thirty seconds
The words, first
The idea: Four forms, and the one fact each one hands you for free.
| Word | What it means |
|---|---|
| Standard form | y = ax² + bx + c. Hands you the y-intercept, c, and the direction from a. |
| Vertex form | y = a(x − p)² + q. Hands you the vertex, (p, q), and therefore the maximum or minimum. |
| Factored form | y = a(x − r₁)(x − r₂). Hands you the x-intercepts, r₁ and r₂. |
| Vertex | The turning point. The answer to every maximum or minimum question. |
| Root | A value of x making y zero — an x-intercept. Also called a zero. |
| Discriminant | b² − 4ac. Tells you how many roots there are before you look for them. |
| Axis of symmetry | x = p, the vertical line through the vertex. Also x = −b/(2a), and the midpoint of the two roots. |
What each form is good for
The idea: Read the question, decide which fact you need, and go to the form that has it.
| Question | Best form | Why |
|---|---|---|
| What is the maximum? | Vertex | It is q, written down |
| Where does it cross the x-axis? | Factored | The roots are in the brackets |
| Where does it cross the y-axis? | Standard | It is c |
| How many roots are there? | Standard | Compute b² − 4ac |
| Is it wide or narrow, up or down? | Any | a is the same in all three |
The axis of symmetry is the bridge. It is x = p in vertex form, x = −b/(2a) in standard form, and the midpoint of the roots in factored form. If you know any two roots, the vertex is halfway between them, and one substitution gives the rest.
Moving between forms
The idea: Three conversions. Two are mechanical; the third is completing the square, which is the only one worth practising.
Vertex → standard. Expand. y = 2(x − 3)² + 1 = 2(x² − 6x + 9) + 1 = 2x² − 12x + 19.
Factored → standard. Expand. y = (x − 1)(x + 5) = x² + 4x − 5.
Standard → vertex: complete the square. Take half the coefficient of x, square it, add and subtract it.
y = 2x² − 8x + 3 = 2(x² − 4x) + 3 = 2(x² − 4x + 4 − 4) + 3 = 2(x − 2)² − 8 + 3 = 2(x − 2)² − 5
The −4 gets multiplied by the 2 on its way out of the bracket. That step is where almost every error in this conversion happens, and checking with x = 0 catches it: 2(4) − 5 = 3 ✓, which matches the original constant.
Standard → factored. Factor if you can; otherwise find the roots with the quadratic formula and build the brackets from them.
Solving a real question end to end
The idea: The method is always: name the quantity being maximised, write it as a quadratic, find the vertex, then answer the question that was actually asked.
A rectangular pen against a wall, with 40 m of fence for three sides.
- Let the two equal sides be x. Then the third side is 40 − 2x.
- Area: A = x(40 − 2x) = −2x² + 40x.
- Vertex: x = −b/(2a) = −40/(−4) = 10.
- Maximum area: A = −2(100) + 400 = 200 m², with sides 10 m, 10 m and 20 m.
Note that the answer is not a square this time, because one side is free. It is worth noticing when a familiar result stops applying.
A projectile. h = −5t² + 20t + 1. Vertex at t = 2 s, giving h = 21 m. If the question asks when, say 2 seconds; if it asks how high, say 21 metres. Reporting p when the question wanted q is a whole-mark error and an extremely common one.
Sanity checks worth thirty seconds
The idea: Four quick tests that catch most arithmetic errors before they are marked.
- Substitute x = 0 into your converted form and check it gives the original c.
- Does the vertex sit between the roots? It has to, and exactly halfway.
- Does a still match? Converting never changes a.
- Is the answer physically sensible? A negative length, a negative time or a maximum below the starting height means something went wrong.