Quadratic Functions and
Parabolas
Linear or Not?
Month
May
June
July
Aug
Sep
Oct
Nov
Dec
Avg Temp
64
67
71
72
71
67
62
58
Avg Temp
Define the variables:
T(m) = Avg monthly temp, in
degrees F, in San Diego,
California.
m = month of the year; ex: m =
5 represents may
TI –84
STAT button then 1:Edit enter
The screen will be L1.
Enter the values for x (Independent values) for L1.
In the L2 column enter the values for y (Dependent
Values) for L2.
Then WINDOW button
Adjust the x min to be lower than the given domain and
the max is higher than the max value.
Adjust the y min to be lower than min given value and
higher than max value.
Graph
Quadratic Functions
Non linear functions that have a squared term and cannot
be written in the general form of a line.
Represented 2 ways
Standard: f(x) = ax2 + bx + c
Vertex: f(x) = a(x – h) 2 + k
Parabolas

A ‘U’ shape or an upside down ‘U’
Vertex
Point where the lowest or minimum
value occurs
Point where the maximum value occurs.
Quadratic Function models this
pattern.
Reading a quadratic
Reading a Quadratic

For what x-values is the curve increasing?

Decreasing

Vertex?

Y-intercept

X-intercept

f(1)
Intercepts

Quadratic can have two xintercepts, one x-intercept or no xintercept

Can have one y- intercept or no yintercept.
Graphing Quadratics in Vertex
form
Graph
the following on
the calculator:
f(x) = x2
f(x) = x2 + 2
f(x) = x2 + 5
Graphing Quadratics: TI 84
Graph
the following
2
f(x) = x
f(x) = x2 - 2
f(x) = x2 - 5
Graphing Quadratics in Vertex
form

f(x) = x2

f(x) = (x + 2) 2

f(x) = (x + 5) 2

f(x) = x2

f(x) = (x - 2) 2

f(x) = (x - 5) 2
Vertex

The vertex in Vertex form:
F(x) = a(x – h)2 + k
Vertex: (h, k)
Symmetry about the vertical line through
the vertex: Axis of Symmetry
x=h
Practice
f(x) = (x + 3)2 + 5
Increasing/Decreasing?
Find the vertex
Find the axis of symmetry
What is the vertical intercept
Sketch the graph
Application
Number of applicants for asylum in the
UK in thousands can be modeled by
A(t) = -3.5(t - 11)2 + 81
Where A(t) represents the number of
applicants for asylum in thousands t
years since 1990.
a) How many applicants applied in 2003?
b) Sketch a graph of this model
c) Give a reasonable domain and range

Domain and range

Domain and range for Quadratics
application are restricted.

Model breakdown occurs if the spread is
too wide.

Therefore to determine the domain and
range rely on the information to make an
informed decision.
Quadratic Function

Domain for quad functions that are
not applications has no restriction

Range is dependent on the vertex of
the graph.
facing upward [k, 00)
facing downward (-oo, k]
Where the vertex is (h, k)
What is the domain and
range?
f(x)
= 2(x + 7)2 + 4
g(x)
= -0.3(x – 2.7)2 – 8.6
Quadratic Model
Avg temp in Charlotte NC.
Month
Temp (oF)
Mar
62
Apr
72
May
80
Jun
86
July
89
Aug
88
Sep
82
Oct
72
Nov
62
Model

Find an equation for a model of these data

Using the model estimate the temp in Dec

The actual avg high temp in Dec for Charolette is 53oF.
How well does your model predict this value?