MCR3U
GOAL:
Exploring the Properties of Quadratic Functions
Name:
To identify and examine the properties of quadratic functions in algebraic and
graphical forms.
Minds On
Use the internet to find videos and/or tutorial sources to review “what you already know”
about quadratic functions. Consider the questions below to help you get started. Brainstorm
and write your ideas in point form in your notes.
•
•
•
What does the equation look like? Is there more than one way of writing a quadratic
equation?
What does the graph look like? What is the graph called?
What features are present in the graph that help me recognize a quadratic function?
Understanding the Graphical Form of a Quadratic Function
The graph of a quadratic function is called a
.
There are several important features of the graph of a quadratic function. Use the graph below
to identify and define each feature of the graph.
The Step-Pattern of a Quadratic Function
To identify the step-pattern of a quadratic function we determine the 1st differences in y as x
changes by one unit. We will examine this for
.
x
-4
-3
-2
-1
0
1
2
3
4
y
16
9
4
1
0
1
4
9
16
1st Differences in y
7
2nd Differences in y
2
5
3
2
1
2
1
2
3
2
5
2
If the 2nd Differences are a constant, this tell us the relationship is quadratic.
7
2
The 1st differences tell us the step pattern. We can see that the ratio of the step pattern for
the function
, moving way from the vertex is
,
,
,
To recognize a quadratic relation from a table of values we always find the 1st differences in
y. We must recognize equivalent ratios of the step-pattern as well. To find the multiplier on
the step pattern we determine what factor the step pattern has been multiplied by.
Example
Determine if each of the following relations are quadratic.
A.
x
-4
y
32
-3
18
-2
-1
0
1
2
3
4
8
2
0
2
8
18
32
1st Differences in y
2nd Differences in y
,
The differences in y follow the step pattern
a is
. Therefore the relation is
,
,
, ..., and the multiplier
.
B.
x
-4
-3
-2
-1
0
1
2
3
4
y
-80
-45
-20
-5
0
-5
-20
-45
-80
1st Differences in y
2nd Differences in y
The differences in y follow the step pattern
a is
. Therefore the relation is
,
,
,
.
, ..., and the multiplier
Understanding the Algebraic Forms of Quadratic Functions
There are three forms of the quadratic function. Each form tells us different information about the
graph of the function.
Vertex Form
Standard Form
Factored Form
We will now examine the graphs of several quadratic functions in each form, and compare each
graph and its features to its equation. We will then determine how the equation can tell us certain
information about the graph.
, ...
A.
Investigating Vertex Form
Use the graphing calculator on the Ipad to graph the following functions. For each graph identify
all key features (axis of symmetry, x-intercepts, y-intercept, vertex).
Features
Features
Features
Examine the features you identified in each graph. Compare these features to its equation.
What features could be predicted or identified from the equation?
B.
Investigating Standard Form
Use the graphing calculator on the Ipad to graph the following functions. For each graph identify
all key features (axis of symmetry, x-intercepts, y-intercept, vertex).
Features
Features
Features
Examine the features you identified in each graph. Compare these features to its equation.
What features could be predicted or identified from the equation?
C.
Investigating Factored Form
Use the graphing calculator on the Ipad to graph the following functions. For each graph identify
all key features (axis of symmetry, x-intercepts, y-intercept, vertex).
Features
Features
Features
Examine the features you identified in each graph. Compare these features to its equation.
What features could be predicted or identified from the equation?
Consolidation of Concepts
1.
List the four important features of the quadratic function graph and explain what they tell
you about the function.
2.
Fill in the flow chart below to summarize the three algebraic forms of the quadratic function
and what each form tells us about the graph of the function.
Vertex Form
Standard Form
Factored Form