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12345Quadratic functions

Solutions to the exercises

Exercise 1
a

First you shift -8 units in the x -direction, then you multiply by 2 along the x -axis and finally you shift -8 units in the y -direction.

b

If you swap the last two steps around then you get h ( x ) = 2 ( ( x + 8 ) 2 - 8 ) = 2 ( x + 8 ) 2 - 16 .

Exercise 2
a

Max. f ( -4 ) = 5

b

- 2 ( x + 4 ) 2 = - 10 gives you ( x + 4 ) 2 = 5 and therefore x = - 4 ± ( 5 ) .

c

-2 ( x + 4 ) 2 + 5 = 5 gives you ( x + 4 ) 2 = 0 and therefore x = -4 .

d

-2 ( x + 4 ) 2 + 5 = -10 gives you ( x + 4 ) 2 = 7,5 and therefore x = -4 ± 7,5 .

e

-2 ( x + 4 ) 2 + 5 = -3 gives you ( x + 4 ) 2 = 4 and therefore x = -2 x = -6 . Solution: -6 < x < -2 .

f

-2 ( x + 4 ) 2 + 5 = 0 gives you ( x + 4 ) 2 = 2,5 and therefore x = -4 ± 2,5 .
Solution: x < -4 - 2,5 x > -4 + 2,5 .

g

-2 ( x + 4 ) 2 + 5 = 20 gives you ( x + 4 ) 2 = -7,5 so there are no solutions. Solution: this is true for every x .

Exercise 3
a

Vertex ( -2 , 10 ) , open down, decreasing for x > -2 .

b

No, at the vertex the function is neither increasing nor decreasing.

c

-3 ( x + 2 ) 2 + 10 = 0 gives you ( x + 2 ) 2 = 10 3 and therefore x = -2 ± 10 3 .
X-intersects ( -2 - 10 3 , 0 ) en ( -2 + 10 3 , 0 ) .

Exercise 4
a

2 ( x - 1 ) 2 = 3 gives you ( x - 1 ) 2 = 1,5 and x = 1 ± 1,5 .

b

First shift by -2 in the x -direction, then multiply by 2 along the y -axis and finally shift by -3 in the y -direction.

c

Doen.

Exercise 5
a

5 ( x - 1 ) 2 = 13 gives you ( x - 1 ) 2 = 2,6 and therefore x = 1 ± 2,6 .
Solution of the inequality: x < 1 - 2,6 x > 1 + 2,6 .

b

5 - x 2 = -21 gives you x 2 = 26 and therefore x = ± 26 .
Solution of the inequality: - 26 < x < 26 .

c

3 ( x - 1 ) 2 = 40 gives you ( x - 1 ) 2 = 40 3 and therefore x = 1 ± 40 3 .
Solution of the inequality: 1 - 40 3 < x < 1 + 40 3 .

d

-4 ( x + 80 ) 2 = -60 gives you ( x + 80 ) 2 = 15 and therefore x = -80 ± 15 .
Solution of the inequality: -80 - 15 < x < -80 + 15 .

Exercise 6
a

Top ( 5 , 4 ) geeft: h ( x ) = a ( x - 5 ) 2 + 4 .
Grafiek door ( 0 ; 2,5 ) geeft: 25 a + 4 = 2,5 en dus a = -0,06 .
Conclusie: h ( x ) = -0,06 ( x - 5 ) 2 + 4 .

b

h ( x ) = 3,05 gives you: ( x - 5 ) 2 = 95 6 and therefore x = 5 ± 95 6 .
This gives you x 1,02 x 8,98 . The player was about 8,98 m away from the basket.

Exercise 7
a

It is a parabola that opens down, with a maximum value of c for x = 3 .

b

The function g ( x ) = - 1 2 ( x - 3 ) 2 is a parabola that opens down with a vertex on the x -axis. In order to get two intersects, you need a c > 0 , then the vertex will be above the x -axis.

c

c < 4

d

Vertex ( 3 , c ) on y = 4 x - 5 gives you: c = 4 3 - 5 = 7 .

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