2 - Describing the Graph of a Parabola

7/25/2019 2 - Describing the Graph of a Parabola

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Describing the graph of a

ParabolaThere are four different ways to describe the graph of a

parabola:

1. It intersects the x-axis twice.

2. It is tangent to the x-axis. (It only intersects it once

!. It lies entirely abo"e the x-axis.

#. It lies entirely below the x-axis.

$e are going to explore all of these ways by loo%ing at

the discri&inant and the coefficient of the s'uared

ter&.


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$e need to be able to

deter&ine if a parabola opens

up or down. To do that we

loo% at the coefficient of the

s'uared ter&.

56

2

= xxy

Type in the following

e'uation into y) on your

calculator.

*ow type the following

e'uation into y) on your

calculator.

2

43 xxy +=

$hat did we learn+

If the s'uared ter& is positi"e it opens up. If

it is negati"e it opens down,


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$e need to be able to

deter&ine what &a%es a

parabola touch the x-axis and

lie abo"e or below it:

If the roots are real the

parabola touches the x-axis:

352

++= xxyEx:

acb 42

( ) ( ) ( )3145 2

133

5

1

=

=

=

c

b

a

The discri&inant is a positi"e

non-perfect s'uare.

Therefore the roots are:

1. eal

2. Irrational!. ne'ual

$e would describe this graph

as touching the x-axis twice,


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If the roots are i&aginary the

parabola doesn/t touch the x-axis:

852

+=

xxyacb 42

( ) ( ) ( )8145 2

78

5

1

=

=

=

c

b

a

$e would describe this graph

as follows:

It lies entirely below the x-

axis.

The discri&inant is negati"e.


1. I&aginary

If the roots are i&aginary and the coefficient of the s'uared ter& is

negati"e it lies entirely below the x-axis. If the roots are i&aginary and

the s'uared ter& is positi"e it lies entirely abo"e the x-axis.

Opensdown


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*ow lets deter&ine what the graph

loo%s li%e.

Page 2

822

= xxy#13:

8

2

1

=

=

=

c

b

aParabola opens up

acb 42

( ) ( ) ( )8142 2

36


perfect s'uare. Therefore the

roots are:1. eal

2. ational

!. ne'ual

0ince the roots are une'ual and real

the graph intersects the x-axis twice,

62=xy#16:

6

0

1

=

=

=

c

b

aParabola opens up

acb 42

( ) ( )( )6140 2

24


non-perfect s'uare.


1. eal

2. Irrational

!. ne'ual

0ince the roots are une'ual and real

the graph intersects the x-axis twice,


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210xy =#19:

0

0

10

=

=

=

cb

aParabola opens up

acb 42

( ) ( )( )01040 2

0

The discri&inant is ero.Therefore the roots are:

1. eal

2. ational

!. 'ual

0ince the roots are e'ual the

parabola is tangent to the x-axis.

732

+= xxy#20:

7

3

1

=

=

=

cb

aParabola opens down

acb 42

( ) ( ) ( )7143 2

19

The discri&inant is negati"e.Therefore the roots are:

1. I&aginary

0ince the roots are i&aginary and the

parabola opens down then the

parabola lies entirely below the x-

axis.

Page 2


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3!#: The roots of are e'ual when % is:022

=++ kxx

e&e&ber the roots are e'ual

when the discri&inant is 4.

042

= acb

kc

b

a

=

=

=

2

1

022

=++ kxx

( ) ( ) ( ) 0142 2 = k

044 = k

k4+ k4+

k44 =4 4

k=1

Page 2


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3!5: The roots of are i&aginary when b is:082

=++ bxx

e&e&ber the roots are i&aginary

when the discri&inant is negati"e.

042


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3!7: If the graph of is tangent to the x-

axis then the roots of are:cbxaxy ++= 2

If a graph is tangent to the x-axis it

only touches the x-axis8888888888.

cbxaxy ++= 2

9*

The roots are:

ealational'ual

Page 2


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3!5: The roots of are e'ual when b is:0162

=++bxx

e&e&ber the roots are e'ual

when the discri&inant is ero.

042

= acb

16

1

=

=

=

c

bb

a

0162

=++bxx

( ) ( ) ( ) 016142 =b

0642

=b

) 4 &eans e'ual roots.

( ) ( ) 088 =+ bb

08 =b

8=b

08 =+b

8=b

Page 2


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;ind the largest integral "alue of %

such that the roots of the gi"en

e'uation are real.

06:42#

2

=++ kxx

kc

b

a

=

=

=

6

104

2 acb

means

real.

( ) ( ) ( ) 0146 2

k

0436 k

k4+ k4+

k436

4 4

k9

9k


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2 - Describing the Graph of a Parabola