What is a quadratic function?
A function is a rule that gives one output y for each input x.
A quadratic function has the rule y = ax² + bx + c. Here a, b and c are fixed numbers and a ≠ 0. The highest power of x is 2.
You can spot a quadratic in a table too. For x = 0, 1, 2, 3, 4 the y-values of y = x² are 0, 1, 4, 9, 16. The first differences are 1, 3, 5, 7. The second differences are all 2. Equal second differences mean the pattern is quadratic.
The graph of every quadratic function is a parabola. It is a smooth U or ∩ shape. It is never a straight line and never has sharp corners.
Key features of a parabola
- Direction: a > 0 opens up (U); a < 0 opens down (∩).
- Width: a bigger |a| gives a narrower curve; a smaller |a| gives a wider one.
- Vertex: the turning point. It is the lowest point (minimum) when a > 0 and the highest point (maximum) when a < 0.
- Axis of symmetry: the vertical line through the vertex, x = −b/2a. The two halves are mirror images.
- y-intercept: put x = 0, you get y = c.
- x-intercepts (roots or zeros): where y = 0. A parabola can cut the x-axis in 2 points, touch it at 1 point, or miss it.
- Domain is all real numbers. Range is y ≥ k if a > 0, and y ≤ k if a < 0.
Transformations and vertex form
Start with the parent graph y = x². Every other parabola is this one stretched, flipped or moved.
Vertex form: y = a(x − h)² + k.
- a stretches (|a| > 1), squashes (|a| < 1) or flips (a < 0) the curve.
- h moves it right by h (left if h is negative). Careful: y = (x − 3)² moves right 3.
- k moves it up by k (down if k is negative).
So the vertex is (h, k) and the axis is x = h.
From standard to vertex form (completing the square): y = x² − 6x + 5 = (x² − 6x + 9) − 9 + 5 = (x − 3)² − 4. Vertex (3, −4).
Factored form y = a(x − p)(x − q) shows the roots p and q directly. The axis is halfway: x = (p + q)/2.
Quadratic functions and equations
Solving ax² + bx + c = 0 is the same as asking: where does the parabola cross the x-axis?
- Discriminant D = b² − 4ac > 0: two crossing points.
- D = 0: the vertex just touches the axis (one repeated root).
- D < 0: no crossing, no real roots.
If the roots are not neat, read approximate roots from the graph, or narrow them down with a table: if y changes sign between x = 1 and x = 2, a root lies between them.
Parabola and a line: to see where y = x² meets y = x + 2, set x² = x + 2, so x² − x − 2 = 0, giving x = 2 or x = −1. Two meeting points. If the new equation has D = 0 the line only touches (a tangent); if D < 0 they never meet.
Maximum and minimum problems
Many real questions ask for "the most" or "the least". If the quantity is a quadratic, the answer is at the vertex.
- Write the quantity as a quadratic in one variable.
- Find the vertex with x = −b/2a (or vertex form).
- Check it makes sense (length positive, time inside the allowed range).
Example: 40 m of fence makes a rectangle. If one side is x, the other is 20 − x. Area A = x(20 − x) = −x² + 20x. Vertex at x = −20/(2·(−1)) = 10. Largest area = 10 × 10 = 100 m². The best rectangle is a square.
Try it: a paper-ball parabola
Throw a paper ball gently across a room in front of a wall. Ask a friend to film it on a phone. Pause the video every few frames and mark the ball on the screen with a pen or a sticky note. Join the marks: you get a ∩-shaped parabola. The highest mark is the vertex. Now open the 3D above, set a negative a and try to match your curve.
Key formulas and definitions
- y = ax² + bx + c (a ≠ 0)
- Vertex form: y = a(x − h)² + k, vertex (h, k)
- Axis of symmetry: x = −b / 2a
- Factored form: y = a(x − p)(x − q), roots p and q
- D = b² − 4ac decides 2, 1 or 0 x-intercepts
Worked examples
1. Say whether y = −2x² + 3 opens up or down and give its vertex.
a = −2 is negative, so it opens down (∩). There is no bx term, so b = 0 and the vertex is at x = 0, y = 3. Vertex (0, 3), a maximum.
2. Write the vertex, axis and range of y = 3(x + 1)² − 5.
Compare with a(x − h)² + k: h = −1, k = −5. Vertex (−1, −5). Axis x = −1. a = 3 > 0, so the range is y ≥ −5.
3. Find the vertex of y = x² − 4x + 3 and sketch it.
a = 1, b = −4. x = −b/2a = 4/2 = 2. y = 4 − 8 + 3 = −1. Vertex (2, −1). y-intercept 3. Roots: (x − 1)(x − 3) = 0, so x = 1 and 3. Plot these five facts and draw a U through them.
4. Write y = 2x² + 8x + 5 in vertex form.
Take 2 out of the first two terms: y = 2(x² + 4x) + 5. Complete the square: x² + 4x = (x + 2)² − 4. So y = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3. Vertex (−2, −3).
5. A ball's height is h = 20t − 5t² metres after t seconds. Find the highest point and when it lands.
a = −5, b = 20. t = −20/(2 × −5) = 2 s. h(2) = 40 − 20 = 20 m, the highest point. Landing: h = 0 → 5t(4 − t) = 0 → t = 0 or t = 4. It lands after 4 s.
6. A shop sells x bags a day at a profit P = −x² + 30x − 125 rupees. How many bags give the most profit?
a = −1, b = 30. x = −30/(−2) = 15 bags. P(15) = −225 + 450 − 125 = ₹100. Selling 15 bags gives the most profit, ₹100.
7. Where does y = x² meet the line y = 2x + 3?
Set x² = 2x + 3 → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0. x = 3 gives y = 9; x = −1 gives y = 1. Meeting points (3, 9) and (−1, 1).
Common mistakes
- Reading y = (x − 3)² as a shift to the left. The minus inside the bracket moves the graph right by 3.
- Using x = b/2a instead of x = −b/2a. The minus sign matters: for y = x² − 6x + 1 the axis is x = 3, not −3.
- Forgetting that a negative a gives a maximum, not a minimum.
- Stopping at the x of the vertex. The question often wants the y-value (the biggest area, the highest height) too.