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Quadratic   Functions:   Exercises

1­A Precalculus

Quadratic  functions:  Quadratic  functions:  Exercise 1Exercise 1

Shift the graph of the parabola y = x²   

a) upward by one unit

b)   downward by 2 units

c)   to the right by 4 units

d)   to the left by 2 units.

In each case, write down the equation of the parabola.

1­E Precalculus

Quadratic  functions:  Quadratic  functions:  Solution  1 a,bSolution  1 a,b

a) one unit upward:

b)   2 units downward:

Fig. L1­1:  Graphs of  the  functions

1­1

g1 x = x 2 1

g2 x = x 2 − 2

Precalculus

c)  4 units to the right:

d)  2 units to the left:

1­2

g1 x = x − 4 2

g2 x = x 2 2

Quadratic  functions:  Quadratic  functions:  Solution  1 c,dSolution  1 c,d

Fig. L1­2:  Graphs of  the  functions

Precalculus

Exercise 2:

Which shifts of the parabola y = x² are the followingfunctions corresponding to ?

Exercise 3:

Which shifts of the graph of the parabola y = x² in the direction of the coordinate axes lead to thefollowing equations:

In all cases, determine the coordinates of the vertex.

2­E

a ) y = x − 2 2 1

b ) y = x 1 2 − 4

c ) y = x − 3 2 − 4

a ) y = x 2 − 8 x 7

b ) y = x 2 4 x 3

c ) y = x 2 − x 12

Quadratic  functions:  Quadratic  functions:  Exercises  2,3Exercises  2,3

Precalculus

2­1a

g1 x = x2 g2 x = x − 22 f x = x − 22 1

Quadratic  functions:  Quadratic  functions:  Solution  2 aSolution  2 a

Fig. L2a:  Graphs of  the  functions

Precalculus

g1 x = x2 g2 x = x 12 f x = x 12 − 4

2­1b

Quadratic  functions:  Quadratic  functions:  Solution  2 bSolution  2 b

Fig. L2b:  Graphs of  the  functions

Precalculus

2­1c

g1 x = x2 g2 x = x − 32 f x = x − 32 − 4

Quadratic  functions:  Quadratic  functions:  Solution  2 cSolution  2 c

Fig. L2c:  Graphs of  the  functions

Precalculus

2 )    4 units to right:

3 )    9 units down:

2­2

a ) y = x2 − 8 x 7 = x2 − 2⋅ 4 x 7 =

= x2 − 2⋅ 4 x 16 − 16 7 =

= [ x2 − 2⋅ 4 x 16] [7 − 16 ] = x − 42 − 9

1 ) y = x2

y = x − 42

y = x − 42 − 9, S 4, −9

b ) y = x2 4 x 3 = x 22 − 1, S −2, −1

c ) y = x2 − x 12= x − 1

2 2

14, S 1

2,

14

Quadratic  functions:  Quadratic  functions:  Solution  3Solution  3

Precalculus

3­A

Determine the vertices of the following functions

y = a x − m2 n

Draw these functions according to

g1 x = a x2 g2 x = a x − m2

f x = a x − m2 n

a ) f x = x 2 − 4 x 5

by transformation to equations like

b ) f x = x 2 6 x 7

c ) f x = x 2 4 x 6

d ) f x =x 2

2 x

72

e ) f x = −2 x 2 − 4 x

f ) f x = −x 2

2− x

12

Quadratic  functions:  Quadratic  functions:  Exercise  4Exercise  4

Precalculus

3­1

g1 x = x 2 , g 2 x = x − 2 2 , f x = x − 2 2 1

y = x 2 − 4 x 5 = x − 2 2 1, S = 2, 1

Fig.  L4­1:  Quadratic  function  y = f (x),  vertex  S

Quadratic  functions:  Quadratic  functions:  Solution  4aSolution  4a

Precalculus

3­2g1 x = x 2 , g 2 x = x 3 2 , f x = x 3 2 − 2

y = x 2 6 x 7 = x 3 2 − 2, S = −3, −2

Quadratic  functions:  Quadratic  functions:  Solution  4bSolution  4b

Fig.  L4­2:  Quadratic  function  y = f (x),  vertex  S

Precalculus

3­3

g1 x = x 2 , g 2 x = x 2 2 , f x = x 2 2 2

y = x 2 4 x 6 = x 2 2 2, S = −2, 2

Quadratic  functions:  Quadratic  functions:  Solution  4cSolution  4c

Fig.  L4­3:  Quadratic  function  y = f (x),  vertex  S

Precalculus

3­4g1 x =

x 2

2, g2 x =

12 x 1 2 , f x =

12 x 1 2 3

y =x 2

2 x

72=

12x 1 2 3, S = −1, 3

Quadratic  functions:  Quadratic  functions:  Solution  4dSolution  4d

Fig.  L4­4:  Quadratic  function  y = f (x),  vertex  S

Precalculus

3­5g1 x = −2 x 2 , g 2 x = −2 x 1 2 , f x = −2 x 1 2 2

y = −2 x 2 − 4 x = −2 x 1 2 2, S = −1, 2

Fig.  L4­5:  Quadratic  function  y = f (x),  vertex  S

Quadratic  functions:  Quadratic  functions:  Solution  4eSolution  4e

Precalculus

3­6g1 x = −

x 2

2, g 2 x = −

12 x 1 2 , f x = −

12x 1 2 1

y = −x 2

2− x

12= −

12

x 1 2 1, S = −1, 1

Quadratic  functions:  Quadratic  functions:  Solution  4fSolution  4f

Fig.  L4­6:  Quadratic  function  y = f (x),  vertex  S

Precalculus