The graph of a quadratic function y = ax² + bx + c is a smooth, symmetric curve called a parabola.
It is the very shape a ball traces when you toss it into the air.
The sign of a decides whether the curve opens upward or downward, and a larger absolute value makes it narrower.
Here you can drag the sliders for a and the vertex position p, q to watch the parabola widen, flip, and glide across the plane.
y = ax² — Role of a
seegongsik.com
Watch how the parabola changes as a varies
1
📊 What a Determines
①a > 0: opens upward
②a < 0: opens downward
③Larger |a| → narrower; smaller |a| → wider
④a = 0 isn't a quadratic
y = a(x−p)²+q — Vertex Form
seegongsik.com
See how p and q translate the parabola
1
2
-1
🎯 Idea of Translation
①y=ax² → y=a(x−p)²: shifts +p along x-axis
②y=ax² → y=ax²+q: shifts +q along y-axis
③Vertex: (p, q), axis: x = p
④Watch signs: y=a(x−2)²+3 → vertex (2, 3)
Standard → Vertex Form
Standard Form
y = ax² + bx + c
Expanded form — coefficients direct
Vertex Form Conversion
y = a(x + b2a)² − b²−4ac4a
Complete the square to expose the vertex
🔄 Conversion Steps
①Factor a out of x²+bx in ax²+bx+c
②Add and subtract (b/2)²
③Group into a perfect square
④Vertex = (−b/2a, −(b²−4ac)/4a)
Putting the Properties Together
Reading the Graph
Vertex (p,q), axis: x=p, y-intercept: c, direction: sign of a
All info from a quadratic graph
💡 Graph Properties
①Axis of symmetry: x = p (x of vertex)
②y-intercept: set x=0, y=c
③x-intercepts: set y=0 → roots of the quadratic
④Max/min: y of the vertex
Work It Out
Example 1
Find the coordinates of the vertex of y=(x−2)²+3.
1
In vertex form y=a(x−p)²+q, the vertex is (p, q).
y = (x−2)²+3
2
Here p=2, q=3, so the vertex is (2, 3).
vertex (2, 3)
▸ (2, 3)
(x−2) is the x−p form, so p=+2. Mind the sign.
Example 2
Rewrite y=x²−4x+1 in vertex form and find the vertex.
1
Complete the square on x²−4x: x²−4x = (x−2)²−4.
y = (x−2)²−4+1
2
So y=(x−2)²−3, and the vertex is (2, −3).
y = (x−2)²−3 ⇒ vertex (2, −3)
▸ y=(x−2)²−3, vertex (2, −3)
Add and subtract the square of half the x-coefficient: (−2)²=4.
Exam Wrap-up
Core Form
y=a(x−p)²+q — vertex (p,q), axis x=p
Starting point for every quadratic-function problem