Graph of Quadratic Functions of the Form
f(x) = + (x-h)2 + k
Prepared by:
Ansiluz H. BetcoSan Bartolome High
School
Objectives:At the end of the lesson, students should be able to :
a) draw the graph of Quadratic Function of the form f(x) = + (x-h)2 + k using sketch pad
b) observe the effects of changes in h and k in the graph of Quadratic Function
c) Sketch the graph of quadratic functions applying its properties.
Target group : Secondary 3ADuration : 50 minutesMode : Student
centered/group work
Do you know that a stream of water that is projected into the air forms a beautiful symmetrical curve?
The curve is the graph of Quadratic Function
Discovering Mathematics 2A
ReviewIdentify the following parts pointed by arrows
Maximum point
Minimum point
Line of symmetry
Line ofsymmetry
http://jwilson.coe.uga.edu/
Sketch the graphs of the following functions on the same plane using Graphmatica. 1) y = x2
2) y = x2 - 3
3) y = (x-3)2 + 2
4) y = (x+5)2 –2
Procedures on how to use Graphmatica Software
Answer
Group Activity 1
Next
Graphmatica.exe
Procedure on how to use Graphmatica
a) Go to graphmatica interactive software. Enter y=x^2 in the function input area and then click enter. The graph of y = x2 will appear on the sketch area with grid lines.
b) Similarly, draw the graphs of y = x2 - 3, y = (x – 3)2 + 2 and y = ( x+ 5)2 – 2 on the same coordinate plane.
c) Study the graphs, copy and complete the following table. Then consider the graph of y = (x – h)2 + k, where h and k are constants Back
y= (x+5)2 - 2
y = x2
y = x2 -3
y = (x-3)2 +2
Answer to Activity 1
Back
Next
Function Line of Symmetry
Turning Point
Is the turning point maximum or minimum?
1) y = x2
x = 0 (0 , 0) Minimum
2) y = x2-3
3) y = (x-3)2 + 2
4) y = (x+5)2 – 2
5) y = (x-h)2 + k
Group WorkComplete the following table
x = 0
x = 3
x = -5
x = h
(0 , -3)
(3 , 2 )
(-5, -2)
(h, k)
Minimum
Minimum
Minimum
Minimum
Sketch the graphs of the following functions on the same plane using Graphmatica.
1) y = - x2
2) y= -(x + 6)2
3) y = -(x+3)2 + 3
4) y = -(x-4)2 – 2
Answer
Group Activity 2
Next
Graphmatica.exe
Answer to Activity 2
y = -x2
y = -(x+6)2
y = (x+3)2 +3
y = -(x-4)2-2
Next
Back
Function Line of Symmetry
Turning Point
Is the turning point maximum or minimum?
1) y =- x2 x = 0 ( 0 , 0) Maximum
2) y =- x2+7
3) y = - (x+3)2 + 4
4) y= - (x - 4)2 - 1
5) y = - (x-h)2 + k
Group Work Complete the following table .
x = 0
x = -3
x = 4
x = h
(0 , 7)
(-3, 4)
(4, -1)
(h, k )
Maximum
Maximum
Maximum
Maximum
Observation Graph of y = (x-h)2 + k
Graph of y = - (x-h)2 + k
1) Compare the shape of the graphs
2) Opening of the graphs
3) Turning point of the graphs
4) The line of symmetry
Complete the table by writing your observation
It has the sameShape as the graph of y= x2
It has the same Shape as the Graph of y = - x2
It opens upwardIt opens downward
Its minimum pt.Is at the point (h,k)
Its maximum pt.Is at the pt. (h,k )
The line x = h is its line of symmetry
The line x=h is itsLine of symmetry (Discovering Mathematics 3A)
1)Sketch the graph of y = (x-1)2 +2 without using Graphmatica
Solution:
First we gather some information before sketching the graph
y = (x-1)2 +2
y = (x-h)2+k( )h,k
1 , 2vertex
Minimum point
when x = o
y = (0 – 1)2 + 2 = 3
( )0 , 3
Now sketch the graph
1)Sketch the graph of y = (x-1)2 +2 without using Graphmatica
y = (x-1)2 +2
2)Do the graph of y = -(x+4)2 +2 on the same plane.
y = -(x+ 4)2+2
follow the same solution y = - (x - h) 2 + k
h , k( ) vertex
- 4 ,2
Maximum point
if x = -2y = - ( -2 + 4)2 +2 = -2
( - 2, -2 )
Now sketch the graph
2)Do the graph of y = -(x+4)2 +2 on the same plane.
y = (x-1)2 +2
y = - (x +4 ) + 2
Graph of y = + ( x – h )2 + k
any questions ?
Let’s practiceSketch the graph of the following functions
1) y = ( x-2)2 + 5 2) f(x) = -( x+7)2 + 3
3) g(x) = ( x-5)2 - 1 4) h(x) = -( x-2)2 + 5
5) p(x) = ( x+ 3)2 – 3/2
Answer to exercises
g(x) = (x-5)2 -1
h(x) = (x-2)2+5
h(x) = -(x-2)2+5
p(x)=(x+3)2 -3/2
f(x)= -(x+7)2 +3
Home work Determined the function whose graphs are
describe below1) The graph of f(x) = x2 shifted 3 units upward2) The graph of g(x)= -42 shifted 6 units below the
origin3) The graph of h(x)=1/4 x2 shifted 2 units above
the x-axis4) The graph of p(x)=-3x2 shifted ½ unit to the
right5) The graph of d(x)= 2x2 shifted 7 units to the left
References:Mathematics 3A by Chow Wai KeungGraphmatica online interactive software