Complete the following graphic organizer to summarize the roles of a, h and k-look at chapter 4
Investigation – What are the roles of a, h and k? You can use Desmos.com
With a partner and using the Parabola Function graphing tool, investigate the roles of a, h and k in the vertex form of the quadratic
function. Record your findings.
Role of a:
As I increase the value of a (larger than 1), I notice…
Role of h:
As I increase the value of h, I notice…
As I decrease the value of a (smaller than -1), I notice…
As I decrease the value of h ( h becomes negative), I notice…
As I change the value of a between -1 and 1, I notice…
When the value of h is zero, I notice…
y = a(x – h)2 + k
Role of k:
As I increase the value of k , I notice…
As I decrease the value of k (k becomes negative), I notice…
When the value of k is zero, I notice…
Source: Edugains.ca
Other Observations:
Complete the following graphic organizer to summarize the roles of a, h and k-look at chapter 4
Demonstrating understanding of the roles of a, h & k in y = a(x – h)2 + k
Equation
Value
of a
Value
of h
Value
of k
Vertex
(h, k)
y = 3(x - 2)2 + 1
a=7
h=2
k=1
(2, 1)
y = -2(x - 3)2 + 3
y=
1
(x +1)2 +5
2
y = 0.3(x + 2)2 + 15
2
y = − ( x − 4) 2 - 8
3
y = 2x2 + 9
y = -3(x + 5)2
Source: Edugains.ca
Step
Pattern
from Vertex
# of
x-intercepts
None
Transformations Starting from y = x2
•
•
•
Vertical stretch by a factor of 3
Translated 2 units right
Translated 1 unit upwards
Complete the following graphic organizer to summarize the roles of a, h and k-look at chapter 4
Role of a:
Role of h:
Direction of Opening:
• When a is positive, the parabola opens ____ ________.
• When a is negative, the parabola opens____________.
Properties:
• If h > 0, then the graph of y = a(x – h)2 + k is translated
horizontally h units to the _______________.
• If h < 0, then the graph of y = a(x – h)2 + k is translated
horizontally h units to the _____________ .
Shape:
• If a > 1 or a < -1, then the graph of y = a(x – h)2 + k is ___
_________to the graph of y = x2.
• If a is between -1 and 1, then the graph of y = a(x – h)2 + k
______________relative to y = x2.
Relation to the Vertex:
• The value of h is the ________ - coordinate of the vertex.
y = a(x – h)2 + k
Example: y = -2(x – 3)2 + 5
Role of k:
Properties:
• If k > 0, then the graph of y = a(x – h)2 + k is translated
vertically k units ____________.
• If k < 0, then the graph of y = a(x – h)2 + k is translated
vertically K units ___________.
State:
a) Direction of opening:
b) Stretch or Compression:
•
Relation to the Vertex:
The value of k is the _____ - coordinate of the vertex.
The graph has been translated ___________
•
X-Intercepts:
• If k = 0, then the graph has __________ zero (x-intercept).
• If k > 0,a>0 then the graph has ______ zeros (x-intercepts).
• If k < 0, a<0 then the graph has ____ __ zeros (x-intercepts).
• If k>0,a<0 then the graph has ________ (two x-intercepts)
• If k<0,a>0 then the graph has ________ (two x-intercepts)
Source: Edugains.ca
Transformations:
The graph has been translated ___________
c) Coordinates of the vertex:)
d) Number of x-intercepts:
Complete the following graphic organizer to summarize the roles of a, h and k-look at chapter 4
Summarizing the roles of a, h & k in y = a(x – h)2 + k (Solution)
Role of a:
Direction of Opening:
• When a is positive, the parabola opens ____up________.
• When a is negative, the parabola opens______down______.
Shape:
• If a > 1 or a < -1, then the graph of y = a(x – h)2 + k is
vertically stretched relative to the graph of y = x2.
• If a is between -1 and 1, then the graph of y = a(x – h)2 + k
vertically compressed relative to y = x2.
Role of h:
Properties:
• If h > 0, then the graph of y = a(x – h)2 + k is translated
horizontally h units to the _left______________.
• If h < 0, then the graph of y = a(x – h)2 + k is translated
horizontally h units to the __________right_____ .
Relation to the Vertex:
• The value of h is the __x___ - coordinate of the vertex.
Example: y = -2(x – 3)2 + 5
Role of k:
Properties:
• If k > 0, then the graph of y = a(x – h)2 + k is translated
vertically k units _____up_______.
• If k < 0, then the graph of y = a(x – h)2 + k is translated
vertically K units _____down______.
State:
e) Direction of opening: down
f) Stretch or Compression: vertically stretched
•
Relation to the Vertex:
The value of k is the ___y__ - coordinate of the vertex.
The graph has been translated 3 units right
•
X-Intercepts:
• If k = 0, then the graph has ______one____ zero (xintercept).
• If k > 0,a>0 then the graph has ___No___ zeros (xintercepts).
• If k < 0, a<0 then the graph has ___No__ zeros (xintercepts).
• If k>0,a<0 then the graph has two zeros (two x-intercepts)
• If k<0,a>0 then the graph has two zeros (two x-intercepts)
Source: Edugains.ca
Transformations:
The graph has been translated 5 units up
g) Coordinates of the vertex: (3,5)
h) Number of x-intercepts: two x-intercepts
because a<0 and k>0
Complete the following graphic organizer to summarize the roles of a, h and k-look at chapter 4
Source: Edugains.ca