LESSON 6 – FACTORED FORM
𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)
A QUADRATIC RELATION can be expressed 3 ways:
STANDARD FORM 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐


VERTEX FORM 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
𝑎 is the stretch or compression factor.
𝑐 is the initial value


𝑎 is the stretch or compression factor.
(ℎ, 𝑘) is the vertex
FACTORED FORM 𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)



𝑎 is the stretch or compression factor.
The x – intercepts, or zeros are 𝑟 and 𝑠.
Due to the symmetric property of a parabola,
the x-coordinate of the vertex is on the axis
of symmetry, which passes through the midpoint
of the line segment connecting the x-intercepts.
∴ The equation of the axis of symmetry is: 𝑥 =
𝑟+𝑠
∴ Vertex (
Example ①
2
𝑟+𝑠
2
, 𝑦)
Sketch the graph of the quadratic relation 𝑦 = −2(𝑥 − 3)(𝑥 + 1). Label the x-intercepts,
vertex, and axis of symmetry.
Example ②
The predicted flight path of a toy rocket used in a mathematics project is defined
by the relation ℎ = −3(𝑑 − 2)(𝑑 − 12), where d is the horizontal distance, in
metres, from a wall, and h is the height, in metres, above the ground.
a) Sketch a graph of the path of the rocket.
b) How far from the wall is the rocket when
it is launched?
c) How far from the wall is the rocket when
it lands on the ground?
d) What is the maximum height of the rocket,
and how far, horizontally, is it from the wall
at that moment?
Example ③ A bridge is a parabolic arch that is approximately 8m tall and 12m wide.
a) Sketch a graph of the arch with left base
located on the x-axis 3units to the left of
the y-axis. Label the x-intercepts and
vertex.
b) Determine an equation in intercept form to model
the arch.