1

a

$y = x^{2} + x$
Nulpunten:
$x^{2} + x = 0$
$x (x + 1) = 0$
$x = 0$ of $x = ‐ 1$

Snijpunt $y$ -as:
$y = 0^{2} + 0 = 0$ $\to$ $(0,0)$
Symmetrieas:
$x = \frac{0 - 1}{2} = ‐ \frac{1}{2}$
$y = {(‐ \frac{1}{2})}^{2} - \frac{1}{2} = ‐ \frac{1}{4}$
Top $(‐ \frac{1}{2}, ‐ \frac{1}{4})$

$y = x^{2} - 7 x$
Nulpunten:
$x^{2} - 7 x = 0$
$x (x - 7) = 0$
$x = 0$ of $x = 7$

Snijpunt $y$ -as:
$y = 0^{2} - 7 \cdot 0 = 0$ $\to$ $(0,0)$

Symmetrieas:
$x = \frac{0 + 7}{2} = 3 \frac{1}{2}$
$y = {(3 \frac{1}{2})}^{2} - 7 \cdot 3 \frac{1}{2} = ‐ 12 \frac{1}{4}$
Top $(3 \frac{1}{2}, ‐ 12 \frac{1}{4})$

$y = ‐ 3 x^{2}$
Nulpunten:
$‐ 3 x^{2} = 0$
$x = 0$

Snijpunt $y$ -as:
$y = ‐ 3 \cdot 0^{2} = 0$ $\to$ $(0,0)$

Symmetrieas:
$x = \frac{0 + 0}{2} = 0$
$y = ‐ 3 \cdot 0^{2} = 0$
Top $(0,0)$

$y = {(x + 2)}^{2} - 3$
Nulpunten:
${(x + 2)}^{2} - 3 = 0$
${(x + 2)}^{2} = 3$
$x + 2 = \sqrt{3}$ of $x + 2 = ‐ \sqrt{3}$
$x = ‐ 2 + \sqrt{3}$ of $x = ‐ 2 - \sqrt{3}$

Snijpunt $y$ -as:
$y = {(0 + 2)}^{2} - 3 = 1$ $\to$ $(0,1)$

Top $(‐ 2, ‐ 3)$

Symmetrieas:
$x = ‐ 2$

$y = ‐ \frac{1}{2} {(x - 1)}^{2} + 8$
Nulpunten:
$‐ \frac{1}{2} {(x - 1)}^{2} = ‐ 8$
${(x - 1)}^{2} = 16$
$x - 1 = 4$ of $x - 1 = ‐ 4$
$x = 5$ of $x = ‐ 3$

Snijpunt $y$ -as:
$y = ‐ \frac{1}{2} {(0 - 1)}^{2} + 8 = 7 \frac{1}{2}$ $\to$ $(0,7 \frac{1}{2})$

Top $(1,8)$

Symmetrieas:
$x = \frac{‐ 3 + 5}{2} = 1$

b

2

$y = x^{2} + 12 x$
Nulpunten:
$x^{2} + 12 x = 0$
$x (x + 12) = 0$
$x = 0$ of $x = ‐ 12$

Symmetrieas:
$x = \frac{0 - 12}{2} = ‐ 6$
$y = {(‐ 6)}^{2} + 12 \cdot ‐ 6 = ‐ 36$
Top $(‐ 6, ‐ 36)$

$y = 2 x^{2} - 5 x$
Nulpunten:
$2 x^{2} - 5 x = 0$
$2 x (x - 2 \frac{1}{2}) = 0$
$x = 0$ of $x = 2 \frac{1}{2}$

Symmetrieas:
$x = \frac{0 + 2 \frac{1}{2}}{2} = 1 \frac{1}{4}$
$y = 2 \cdot {(1 \frac{1}{4})}^{2} - 5 \cdot 1 \frac{1}{4} = ‐ 3 \frac{1}{8}$
Top $(1 \frac{1}{4}, ‐ 3 \frac{1}{8})$

$y = x^{2} + 3 x + 2$
Nulpunten:
$x^{2} + 3 x + 2 = 0$
$(x + 1) (x + 2) = 0$
$x = ‐ 1$ of $x = ‐ 2$

Symmetrieas:
$x = \frac{‐ 1 - 2}{2} = ‐ 1 \frac{1}{2}$
$y = {(‐ 1 \frac{1}{2})}^{2} + 3 \cdot ‐ 1 \frac{1}{2} + 2 = ‐ \frac{1}{4}$
Top $(‐ 1 \frac{1}{2}, ‐ \frac{1}{4})$

$y = ‐ x^{2} + 4 x + 6$
$‐ x^{2} + 4 x + 6 = 6$
$‐ x^{2} + 4 x = 0$
$‐ x (x - 4) = 0$
$x = 0$ of $x = 4$

Symmetrieas:
$x = \frac{0 + 4}{2} = 2$
$y = ‐ 2^{2} + 4 \cdot 2 + 6 = 10$
Top $(2,10)$

3

a

$y = c x^{2}$
$3 = c \cdot 4^{2}$ (invullen het punt $(4,3)$ )
$3 = 16 c$
$\frac{3}{16} = c$
Vergelijking parabool: $y = \frac{3}{16} x^{2}$

b

$x = 3$ of $x = ‐ 3$ $\Rightarrow$ $y = \frac{3}{16} \cdot 3^{2} = 1 \frac{11}{16}$
Dus $(3,1 \frac{11}{16})$ en $(‐ 3,1 \frac{11}{16})$

4

100 ; 10	5 ; $6 \frac{1}{4}$ ; $x$
18 ; 81 ; $x$	6 ; 36
$42 \frac{1}{4}$ ; $6 \frac{1}{2}$	1 ; 1

5

Linkerkolom:
$14 = x (x - 5)$
$x^{2} - 5 x - 14 = 0$
$(x - 7) (x + 2) = 0$
$x = 7$ of $x = ‐ 2$

$2 x^{2} + x = 5 x + 8$
$2 x^{2} - 4 x - 8 = 0$
$x^{2} - 2 x - 4 = 0$
$x^{2} - 2 x - 4 + 5 = 5$
$x^{2} - 2 x + 1 = 5$
${(x - 1)}^{2} = 5$
$x - 1 = \sqrt{5}$ of $x - 1 = ‐ \sqrt{5}$
$x = 1 + \sqrt{5}$ of $x = 1 - \sqrt{5}$

$25 = 4 {(x + 1)}^{2}$
${(x + 1)}^{2} = 6 \frac{1}{4}$
$x + 1 = 2 \frac{1}{2}$ of $x + 1 = ‐ 2 \frac{1}{2}$
$x = 1 \frac{1}{2}$ of $x = ‐ 3 \frac{1}{2}$

Rechterkolom:
${(x + 1)}^{2} + {(x + 3)}^{2} = 4 x^{2}$
$2 x^{2} + 8 x + 10 = 4 x^{2}$
$2 x^{2} - 8 x - 10 = 0$
$x^{2} - 4 x - 5 = 0$
$(x - 5) (x + 1) = 0$
$x = 5$ of $x = ‐ 1$

$x^{2} - 3 x = 2 x^{2} + x + 1$
$x^{2} + 4 x + 1 = 0$
$x^{2} + 4 x + 1 + 3 = 3$
$x^{2} + 4 x + 4 = 3$
${(x + 2)}^{2} = 3$
$x + 2 = \sqrt{3}$ of $x + 2 = ‐ \sqrt{3}$
$x = ‐ 2 + \sqrt{3}$ of $x = ‐ 2 - \sqrt{3}$

$x^{2} + 5 x + 3 = 0$
$x^{2} + 5 x + 3 + 3 \frac{1}{4} = 3 \frac{1}{4}$
$x^{2} + 5 x + 6 \frac{1}{4} = 3 \frac{1}{4}$
${(x + 2 \frac{1}{2})}^{2} = 3 \frac{1}{4} = \frac{13}{4}$
$x + 2 \frac{1}{2} = \sqrt{\frac{13}{4}} = \frac{1}{2} \sqrt{13}$ of $x + 2 \frac{1}{2} = ‐ \sqrt{\frac{13}{4}} = ‐ \frac{1}{2} \sqrt{13}$
$x = ‐ 2 \frac{1}{2} + \frac{1}{2} \sqrt{13}$ of $x = ‐ 2 \frac{1}{2} - \frac{1}{2} \sqrt{13}$

6

a

Border: $2 x^{2} + 3 \cdot 4 x = 2 x^{2} + 12 x$
Dus $2 x^{2} + 12 x = 4 \cdot 4 = 16$
$2 x^{2} + 12 x - 16 = 0$
$x^{2} + 6 x - 8 = 0$
${(x + 3)}^{2} - 9 - 8 = 0$
${(x + 3)}^{2} = 17$
$x + 3 = \sqrt{17}$ of $x + 3 = ‐ \sqrt{17}$
$x = ‐ 3 + \sqrt{17}$ of $x = ‐ 3 - \sqrt{17}$
Maar $x > 0$ , dus $x = ‐ 3 + \sqrt{17}$

b

$2 (2 x^{2} + 12 x) = 16$
$2 x^{2} + 12 x = 8$
$2 x^{2} + 12 x - 8 = 0$
$x^{2} + 6 x - 4 = 0$
${(x + 3)}^{2} - 9 - 4 = 0$
${(x + 3)}^{2} = 13$
$x + 3 = \sqrt{13}$ of $x + 3 = ‐ \sqrt{13}$
$x = ‐ 3 + \sqrt{13}$ of $x = ‐ 3 - \sqrt{13}$
Maar $x > 0$ , dus $x = ‐ 3 + \sqrt{13}$

7

$3 x^{2} + 10 x + 3 = 0$ $\begin{array}{l} a = 3 \\ b = 10 \\ c = 3 \end{array}} \begin{matrix} D = 100 - 4 \cdot 3 \cdot 3 = 64 \\ \sqrt{D} = \sqrt{64} = 8 \end{matrix}$ $x = \frac{‐ 10 + 8}{6} = ‐ \frac{1}{3}$ of $x = \frac{‐ 10 - 8}{6} = ‐ 3$	$x^{2} - 8 x = ‐ 22$ $x^{2} - 8 x + 22 = 0$ $\begin{array}{l} a = 1 \\ b = ‐ 8 \\ c = 22 \end{array}} D = 64 - 4 \cdot 1 \cdot 22 = ‐ 24$ $D < 0$ , dus geen oplossingen
$2 x^{2} = 5 x - 3$ $2 x^{2} - 5 x + 3 = 0$ $\begin{array}{l} a = 2 \\ b = ‐ 5 \\ c = 3 \end{array}} \begin{matrix} D = 25 - 4 \cdot 2 \cdot 3 = 1 \\ \sqrt{D} = \sqrt{1} = 1 \end{matrix}$ $x = \frac{5 + 1}{4} = 1 \frac{1}{2}$ of $x = \frac{5 - 1}{4} = 1$	$‐ 5 x^{2} + 4 x - \frac{4}{5} = 0$ $\begin{array}{l} a = ‐ 5 \\ b = 4 \\ c = ‐ \frac{4}{5} \end{array}} D = 16 - 4 \cdot ‐ 5 \cdot ‐ \frac{4}{5} = 0$ $x = ‐ \frac{4}{‐ 10} = \frac{2}{5}$

8

a

oppervlakte driehoeken $= 2 \cdot x (8 - x)$

b

$\frac{1}{4}$ deel ; $\frac{1}{4} \cdot 8 \cdot 8 = 16$

c

$2 \cdot x (8 - x) = 16$
$2 x^{2} - 16 x + 16 = 0$
$x^{2} - 8 x + 8 = 0$
$\begin{array}{l} a = 1 \\ b = ‐ 8 \\ c = 8 \end{array}} \begin{matrix} D = 64 - 4 \cdot 1 \cdot 8 = 32 \\ \sqrt{D} = \sqrt{32} = 4 \sqrt{2} \end{matrix}$
$x = \frac{8 + 4 \sqrt{2}}{2} = 4 + 2 \sqrt{2}$ cmof $x = \frac{8 - 4 \sqrt{2}}{2} = 4 - 2 \sqrt{2}$ cm

9

a

hoogte = $x$ , breedte = $x + 3$ , lengte = $x + 4$

b

oppervlakte $= 2 (x (x + 4) + x (x + 3) + (x + 4) (x + 3)) =$
$2 (3 x^{2} + 14 x + 12) = 6 x^{2} + 28 x + 24$

$6 x^{2} + 28 x + 24 = 162$
$6 x^{2} + 28 x - 138 = 0$
$\begin{array}{l} a = 6 \\ b = 28 \\ c = ‐ 138 \end{array}} \begin{matrix} D = 784 - 4 \cdot 6 \cdot ‐ 138 = 4096 \\ \sqrt{D} = \sqrt{4096} = 64 \end{matrix}$
$x = \frac{‐ 28 + 64}{12} = 3$ of $x = \frac{‐ 28 - 64}{12} = ‐ 7 \frac{2}{3}$
Alleen $x = 3$ voldoet, omdat $x > 0$ moet zijn.

10

a

$50 t - 5 t^{2} = 0$
$5 t (10 - t) = 0$
$t = 0$ of $t = 10$
Dus de vlucht duurt $10 - 0 = 10$ sec.

b

De maximale hoogte wordt bereikt na 5 sec.
$h = 50 \cdot 5 - 5 \cdot 5^{2} = 250 - 125 = 125$ m

c

d

$50 t - 5 t^{2} > 113,75$
$0 > 5 t^{2} - 50 t + 113,75$
$t^{2} - 10 t + 22,75 < 0$
$(t - 3,5) (t - 6,5) < 0$
$3,5 < t < 6,5$
Dus tussen de 3,5 en 6,5 sec. is de hoogte van de vuurpijl meer dan 113,75 m.