3-1: Characteristics of Quadratic Functions
Unit 3: Quadratic Functions
MCR3U1: Functions
Introduction
You are currently raising money for this year’s food drive and would like to sell a 4GB iPod nano
to get this money. These iPods were donated to you so you can sell them at any price. Your
job is to determine a what price should you sell these iPods to generate the greatest revenue?
LESSON
: Properties of Quadratic Functions, f(x)=ax2+bx + c
Unit
: Quadratic Functions
PART A: Introduction



Quadratic functions have degree
and produce
when graphed.
Quadratic functions can be represented by quadratic equations in different forms.
Each form gives different information about the function:
(i)
Standard Form: f(x) = ax2+bx + c , a≠0
 This form gives the y-intercept, c.
(ii) Factored Form: f(x) = a(x – s)(x – t) , a≠0
 This form gives the zeros (roots or x-intercepts), x = s and x = t.
(iii) Vertex Form:
f(x) = a(x – h)2 + k , a≠0
 This form gives the vertex (h, k) and the maximum or minimum value of
the function, k, when x = h.
Property
Vertex
Axis of Symmetry
Direction of Opening
Sign of a
Positive, a>0 Negative, a<0
Min/Max y-value
PART B: Writing a Quadratic Equation
Ex. 1: Micha owns a business selling snowboards. She collects the following profit data:
Profits from Snowboard Sales


Profit,
P(x)
(x$10000)
-32
-14
0
10
16
18
16
10
0
-14
1st
Diff.
2nd
Diff.
Quadratic functions represented
as a table of values have
constant
differences.
When 2nd differences are:
+ve: parabola opens
-ve: parabola opens
15
Profit
($10000s)
)
Snowboards
Sold
(x1000)
0
1
2
3
4
5
6
7
8
9
10
5
-10
10
-5
-10
-15
-20
-25
-30
-35
20
30
#Snowboards
(1000s)
1. a) Write an algebraic equation to model Micha’s Profit using the vertex
form of a quadratic function.
b) What information do you need to write the vertex form of the quadratic
function?
2. a) Write an algebraic equation to model Micha’s Profit using the factored
form of a quadratic function.
b) What information do you need to write the factored form of the quadratic
function?
Part C: Determining the Properties of a Quadratic Function
Ex. 2: A window washer tosses a tool to his partner across the street. The
height of the tool above the ground is modeled by the quadratic function,
h(t) = -5t2 + 20t + 25, where h(t) is the height in metres and t is the
time in seconds after the toss.
a)
How high above the ground is the window?
b)
If his partner misses the tool, when will it hit the ground?
c)
If the path of the tool’s height were graphed, where would the axis of
symmetry be?
d)
Determine the domain and range of this function.
Part D: Graphing a Quadratic Function Using the Vertex Form
Ex. 3: Given f(x) = 2(x – 1)2 – 5,
a) state the vertex, axis of symmetry, direction of opening, y-intercept,
domain and range.
b) Graph the function.
c) Find two other points on the parabola to help you graph more accurately.
Homefun: p. 145 # 3-4, 5bcd, 6ab, 8 (find equation in 3 different forms), 9ab, 10-12