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