Section 2.2
Quadratic Functions
Quadratic Functions
 Can
be written in two forms (𝑎 ≠ 0). . .
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 +𝑘
OR
𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Parabola
parabola/pəˈrabələ/
axis of
symmetry
𝑥=3
vertex
(3, -2)
Standard Form
𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐 +𝒌, 𝒂 ≠ 𝟎
Vertex
(𝒉, 𝒌)
Axis of Symmetry
𝒙=𝒉
Opens Up
Opens Down
Minimum
Maximum
𝒂>𝟎
𝒂<𝟎
Tester
 Vertex?
 Max
 Axis
or min?
of Symmetry?
 Open
𝑎
up or down?
positive or negative?
Example 1
𝑓 𝑥 = −3 𝑥 − 4
 vertex:
 max
or min
 opens
 line
up or opens down
of symmetry:
2
+7
Polynomial Form
𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄, 𝒂 ≠ 𝟎
−𝒃
−𝒃
,𝒇
𝟐𝒂
𝟐𝒂
Vertex
Axis of Symmetry
Opens Up
Opens Down
𝒙=
Minimum
Maximum
−𝒃
𝟐𝒂
𝒂>𝟎
𝒂<𝟎
i.e. to find the 𝑥coord.
of
the
−𝑏
vertex, find 2𝑎 .
For 𝑦 , plug what
you get into the
equaion.
Example 2
𝑓 𝑥 = −3𝑥 2 − 12𝑥 + 5
 vertex:
 max
or min
 opens
 line
up or opens down
of symmetry:
Graphing Parabolas
1.
Find the vertex.

Standard Form – (ℎ, 𝑘)

Polynomial Form – find
−𝑏
2𝑎
and plug it into the equation
2.
Find the 𝒙-intercepts by setting the equation equal to
0 and solving.
3.
Find the 𝒚-intercept by plugging in 0 for 𝑥.
4.
Plug in additional 𝒙-values if necessary.
5.
Plot points & connect the dots to form a smooth curve.
Example 3.0
Graph the function. State the axis of symmetry,
domain and range.
𝑦 = 𝑥−1
2
−9
Example 3.1
Graph the function. State the axis of symmetry,
domain and range.
𝑦−3= 𝑥−1
2
Example 4
Graph the function. State the axis of symmetry,
domain and range.
9
1
𝑦= − 𝑥−
4
2
2
Trip Down Memory Lane . . .
 Finding
the x-intercepts involves solving
quadratic equations.
 How
do you solve these?

𝑥2 − 9 = 0

𝑥 2 − 7𝑥 + 10 = 0

3𝑥 2 − 7𝑥 + 2 = 0

𝑥 2 − 𝑥 + 21 = 0
Example 5
Graph the function. State the axis of symmetry,
domain and range.
𝑦 = 𝑥 2 − 2𝑥 − 8
Example 6
Graph the function. State the axis of symmetry,
domain and range.
𝑦 = 𝑥 2 + 4𝑥 − 1
Example 7
Determine without graphing.
𝑓 𝑥 = −2𝑥 2 − 12𝑥 + 3
a.
Does it have a min or a max?
b.
Min or Max:
c.
Domain and Range:
Applications of Quadratics
Two concepts often arise in word problems. . .
Quadratic Concept
Vertex
Examples of Applications
“Maximum height”
“Minimum area”
“when does the object hit
the ground”
𝑥 −intercepts
“when will it be at a height
of zero”
Example 8
a)
How far does it travel?
b)
What is the maximum height?
c)
From what height was it released?
Example 9
Among all pairs of numbers whose sum is 22, find a
pair whose product is as large as possible. What is
the maximum product?
Example 10
Questions???
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