We’ve learned what linear functions are and how they model
perfectly linear or nearly linear data.
We now focus on working with nonlinear data ( curvilinear ), specifically
quadratic functions.
Quadratic Function : f(x) = ax2 + bx + c
represents a quadratic function, where a, b, and c are
real numbers with a ≠ 0. The domain of a quadratic function is all real numbers.
Vertex Form : Another way the quadratic function can
be expressed:
f(x) = a ( x-h )2 + k (note: best form to use when modeling data)
• The graph is a parabola , a U-shaped graph that opens
either upward or downward
• a is the leading coefficient → If a > 0, the graph opens upward, if a < 0, the
graph opens
downward , also a controls the width of the parabola (narrow or wide)
• Vertex (h,k) → the highest point on a parabola that opens downward or the
lowest point
that opens upward is called the vertex
• Axis of symmetry (x = h) → the vertical line passing through the vertex is
called the axis of
symmetry
• Maximum or Minimum y- value on the graph (k)
Ex: 2x2 + 2x + 1
Identifying Quadratic Functions
In order to identify a function as “Quadratic Function”, you must be able to
write in the form
f(x) = ax2 + bx + c {or f(x) = a ( x-h )2 + k }
Ex:
Converting from Quadratic Function Form to Vertex Form
You convert from quadratic function to vertex form by completing the square .
If the quadratic
expression can be written as x2 + kx + (k / 2) 2 (i.e., a
perfect square trinomial) and can be factored
as x2 + kx + (k / 2) 2 = (x + k / 2) 2
Ex:
Converting from Vertex Form to Quadratic Function Form
1. Square the binomial , 2. Distribute , 3. Add like terms
Ex:
Vertex Formula
The vertex of any parabola can be determined. The vertex of the graph of f(x) =
ax2 + bx + c with
a ≠ 0 is the point ( - b / 2a , f( - b / 2a ) ).
Note → the vertex formula can be used to write a quadratic
function in vertex form.
Ex:
Applications and Models
Exs:
