PreCalculus Class Notes QF1 Quadratic Functions and Models
Quadratic Function
Let a, b, and c be constants with a ≠ 0.
A function represented by
f(x) = ax2 + bx + c
is a quadratic function.
The graph of a quadratic function is a parabola—a U shaped graph that opens either upward or
downward. The leading coefficient a controls the width of the parabola. Larger values of |a|
result in a narrower parabola, and smaller values of |a| result in a wider parabola.
opens upward if a is positive (a > 0)
opens downward if a is negative (a < 0)
The highest point on a parabola that opens downward (maximum) and the lowest point on a
parabola that opens upward (minimum) is called the vertex. The vertical line passing through the
vertex is called the axis of symmetry. Its equation will always be
−b
x=
2a
Example 2b
Use the graph of the quadratic function to determine
the sign of the leading coefficient,
its vertex,
and the equation of the axis of symmetry.
Give intervals where the function is increasing
and where it is decreasing.
Vertex Form
f(x) = a(x – h)2 + k
vertex (h, k)
Standard Form
f ( x ) = ax 2 + bx + c
axis of symmetry: x =
Converting Between Forms
Vertex Form
f(x) = a(x – h)2 + k
−b
2a
Standard Form
distribute and combine  complete the square 
f ( x ) = ax 2 + bx + c
Example
2
Write the quadratic function f ( x ) = −2 ( x − 4 ) + 7 in standard form.
Example
Write the formula f(x) = x2 + 10x – 3 in vertex form by completing the square.
Add to both sides method
Add/subtract same side method
Completing the square—watch the parentheses!
y = x 2 + bx + c
y = x 2 + bx
2
+c
2
b b

y=  x +  −   + c
2 2

Example with fraction: Rewrite in vertex form
f ( x ) = x 2 − 3x + 5
Example with a ≠ 1: Rewrite in vertex form.
f ( x ) = 2x 2 − 3x + 5
Vertex Formula
The vertex of the graph of f(x) = ax2 + bx + c with a ≠ 0:
 −b  −b  
 2a , f  2a  



Example 7
Use the vertex formula [f(x) = a(x – h)2 + k] to write f(x) = 3x2 + 12x + 7 in vertex form.