Homework # 11

(1)

Homework # 11

An interesting formula

Inthishomework,wewilllookataformulathatallowstondtheformulaofapolynomial

funtionofthe seonddegree, knowingonlytheoordinates ofitsextremalpointandtwo

speial harateristisof the urve.

In the piture on the right-hand side,

(x ⁰ , y 0 )

âre ^theôordinates ôf^the êxtremal

point, and

r

^and

d

^are ^the ^horizontal ^and

vertial distanes between the two points.

In the piture,

x 0

^and

y 0

^are ^positive, ^and

they ould alsobe negative.

Thenumber

r

îsâlled^the^radius ând

d

^the

deviation.

b

x 0

y 0

r d

Partie A – An example

In this part, wewilltry tond the formulaof aquadratifuntion whoseextremal point

is

(3, 7)

^with ^a ^deviation^of

2

^for^a ^radius ^of

4

^.

1. Find the simplest linearfuntion suh that the image of

3

^is

0

^.

2. Turn the previous funtionintoa quadratifuntion suh that the image of

3

^is

0

^.

3. Dedue a quadratifuntion

g

^suh ^that ^the ^image ^of

3

^is

7

^.

4. (a) Usethe radius and the deviationto nd anotherpointon the urve.

(b) Theordinateofthenewpointshouldbetheimageofitsabsissaunderfuntion

g

^. ^Chek ^that ^it ^is^so ^with ^the ^formula^found ^previously^.

() With only one multipliative oeient, turn the formula of funtion

g

^into

the formula ofa new funtion

f

^that ^works ^for ^the ^two^points.

5. Byusing the radiusand deviationsymmetrially,we get the oordinatesof another

point thatshould beonthe urve. Chek that theformulaworksfor thispointtoo.

6. Expand the formulayoufound for

f

ⁱⁿ^the ^previous ^question.

7. Drawthe variationstable of the funtion

f

^,^and ^then ^its ^sign ^table.

Partie B – The general formula

In this part, wedon't knowthe oordinates

(x ⁰ , y 0 )

^,^the ^radius

r

^nor^the ^deviation

d

^.^We

will tryto nd ageneral formula.

1. Find the simplest linearfuntion suh that the image of

x 0

^is

0

^.

2. Turn the previousfuntion intoaquadratifuntion suh thatthe image of

x 0

^is

0

^.

3. Dedue a quadratifuntion

g

^suh ^that ^the ^image ^of

x 0

^is

y 0

^.

4. (a) Usethe radius and the deviationto nd anotherpointon the urve.

(b) Theordinateofthenewpointshouldbetheimageofitsabsissaunderfuntion

g

^. ^Chek ^that ^it ^is^so ^with ^the ^formula^found ^previously^.

(2)

() Usingthe multipliativeoeient

d

r 2

^,^turn ^the ^formula^of ^funtion

g

^into^the

formulaof a new funtion

f

^that ^works ^for ^the ^two ^points.

5. Byusing the radiusand deviationsymmetrially,we get the oordinatesof another

point thatshould beonthe urve. Chek that theformulaworksfor thispointtoo.

Partie C – Applying the formula

Use the formula to nd the quadrati funtions with the harateristis show below. In

eah ase, give the initialexpression, then the expanded one and the variationstable.

1.

x 0 = −2

^,

y 0 = 4

^,

r = 4

^and

d = −2

^.

2.

x 0 = 7

^,

y 0 = − 2

^,

r = 5

^and

d = 3

^.

3.

x 0 = 4

^,

y 0 = 6

^,

r = 3

^and

d = − 5

^.

Homework # 11

Homework # 11

(x 0 , y 0 )

r

d

x 0

y 0

r

d

b

b

x 0

y 0

r d

Partie A – An example

(3, 7)

2

4

3

0

3

0

g

3

7

g

g

f

f

f

Partie B – The general formula

(x 0 , y 0 )

r

d

x 0

0

x 0

0

g

x 0

y 0

g

d

r 2

g

f

Partie C – Applying the formula

x 0 = −2

y 0 = 4

r = 4

d = −2

x 0 = 7

y 0 = − 2

r = 5

d = 3

x 0 = 4

y 0 = 6

r = 3

d = − 5

(x ⁰ , y 0 )

(x ⁰ , y 0 )