Quadratic Functions
“A quadratic is a parabola; its vertex is either the peak or the bottom.”
The formula
y = a(x − p)² + qHow to read it: a parabola with vertex (p, q)
- a
- — the direction it opens (sign) and its width (size)
- (p, q)
- — the parabola's vertex — the max or min point
- x
- — the input value
The hook
If a line is a linear function, a quadratic is a parabola — the very curve a thrown ball traces.
In plain words
y = a(x−p)² + q. The graph is a parabola, and the vertex (p, q) is its 'pointed tip' (the highest or lowest point).
The intuition
If a>0 it's an upward cup (∪), so the vertex is the 'lowest point'; if a<0 it's a downward cap (∩), so the vertex is the 'highest point'. The larger |a|, the narrower and steeper it is.
How it's built
The form y=ax²+bx+c, rearranged by completing the square, becomes y=a(x−p)²+q, revealing the vertex (p,q). p is the position of the axis of symmetry, q is the height there.
Example
y=x²−2x−3 = (x−1)²−4. Vertex (1,−4); since a=1>0, it's an upward parabola with its lowest point at (1,−4). The y-intercept is −3 (at x=0).
Common mistake
Miss the sign of a and you swap max for min — a>0 gives a minimum (a bottom), a<0 gives a maximum (a peak). The vertex's x-coordinate is also tricky, since (x−p) uses p.
Where it's used
Parabolic antennas, a fountain's arc, maximum-area and maximum-profit problems — the moment you ask 'where is it largest or smallest', a quadratic answers.
Where it came from
The parabola shook hands with physics when Galileo showed that a thrown object's path is exactly this curve.
Prerequisites
Quick check
What is the vertex of y = (x − 2)² + 3?
- (2, 3)✓
- (−2, 3)
- (2, −3)
- (3, 2)
Practice
For y = x², what is y when x = 3?
Answer: 9
- y = 3²
- = 9
Takeaway: A quadratic too gives one y for each x.
What is the x-coordinate of the vertex of y = (x − 1)² − 4?
Answer: 1
- In the vertex form a(x−p)²+q, p is the x-coordinate
- (x − 1)² gives p = 1
Takeaway: In (x − p), p is the vertex's x-coordinate.
What is the x-coordinate of the vertex of y = x² − 4x + 3?
Answer: 2
- Axis of symmetry x = −b/(2a) = −(−4)/(2·1)
- = 4/2 = 2
Takeaway: The vertex's x-coordinate comes straight from −b/2a.
Rewrite y = x² − 4x + 3 by completing the square and find the vertex.
Answer: undefined
- x² − 4x = (x − 2)² − 4
- y = (x − 2)² − 4 + 3 = (x − 2)² − 1
- vertex (2, −1)
Takeaway: Completing the square reveals the vertex immediately.