Övningar 2.3

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(Ny sida: __NOTOC__ ==&Ouml;vning 2.3:1== <div class="ovning"> Kvadratkomplettera f&ouml;ljande uttryck <table width="100%" cellspacing="10px"> <tr align="left"> <td class="ntext">a)</td> <td class="...)
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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==&Ouml;vning 2.3:1==+'''&Ouml;vning 2.3:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Kvadratkomplettera f&ouml;ljande uttryck Kvadratkomplettera f&ouml;ljande uttryck
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$x^2-2x$</td> <td class="ntext" width="25%">$x^2-2x$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$x^2+2x-1$</td> <td class="ntext" width="25%">$x^2+2x-1$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$5+2x-x^2$</td> <td class="ntext" width="25%">$5+2x-x^2$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$x^2+5x+3$</td> <td class="ntext" width="25%">$x^2+5x+3$</td>
</tr> </tr>
Rad 18: Rad 18:
</div> </div>
-==&Ouml;vning 2.3:2==+'''&Ouml;vning 2.3:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
L&ouml;s f&ouml;ljande andragradsekvationer med kvadratkomplettering L&ouml;s f&ouml;ljande andragradsekvationer med kvadratkomplettering
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$x^2-4x+3=0$</td> <td class="ntext" width="33%">$x^2-4x+3=0$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$y^2+2y-15=0$</td> <td class="ntext" width="33%">$y^2+2y-15=0$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$y^2+3y+4=0$</td> <td class="ntext" width="33%">$y^2+3y+4=0$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$4x^2-28x+13=0$</td> <td class="ntext" width="33%">$4x^2-28x+13=0$</td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$5x^2+2x-3=0$</td> <td class="ntext" width="33%">$5x^2+2x-3=0$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$3x^2-10x+8=0$</td> <td class="ntext" width="33%">$3x^2-10x+8=0$</td>
</tr> </tr>
Rad 42: Rad 42:
</div> </div>
-==&Ouml;vning 2.3:3==+'''&Ouml;vning 2.3:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
L&ouml;s f&ouml;ljande ekvationer direkt L&ouml;s f&ouml;ljande ekvationer direkt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$x(x+3)=0$</td> <td class="ntext" width="50%">$x(x+3)=0$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$(x-3)(x+5)=0$</td> <td class="ntext" width="50%">$(x-3)(x+5)=0$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$5(3x-2)(x+8)=0$</td> <td class="ntext" width="50%">$5(3x-2)(x+8)=0$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$x(x+3)-x(2x-9)=0$</td> <td class="ntext" width="50%">$x(x+3)-x(2x-9)=0$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$(x+3)(x-1)-(x+3)(2x-9)=0$</td> <td class="ntext" width="50%">$(x+3)(x-1)-(x+3)(2x-9)=0$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$x(x^2-2x)+x(2-x)=0$</td> <td class="ntext" width="50%">$x(x^2-2x)+x(2-x)=0$</td>
</tr> </tr>
Rad 68: Rad 68:
</div> </div>
-==&Ouml;vning 2.3:4==+'''&Ouml;vning 2.3:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m en andragradsekvation som har r&ouml;tterna Best&auml;m en andragradsekvation som har r&ouml;tterna
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">$-1\ $ och $\ 2$</td> <td class="ntext" width="100%">$-1\ $ och $\ 2$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">$1+\sqrt{3}\ $ och $\ 1-\sqrt{3}$</td> <td class="ntext" width="100%">$1+\sqrt{3}\ $ och $\ 1-\sqrt{3}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">$3\ $ och $\ \sqrt{3}$</td> <td class="ntext" width="100%">$3\ $ och $\ \sqrt{3}$</td>
</tr> </tr>
Rad 88: Rad 88:
</div> </div>
-==&Ouml;vning 2.3:5==+'''&Ouml;vning 2.3:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">Best&auml;m en andragradsekvation som bara har $\,-7\,$ som rot.</td> <td class="ntext" width="100%">Best&auml;m en andragradsekvation som bara har $\,-7\,$ som rot.</td>
</tr> </tr>
<tr> <tr>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">Best&auml;m ett v&auml;rde p&aring; $\,x\,$ som g&ouml;r att uttrycket $\,4x^2-28x+48\,$ &auml;r negativt.</td> <td class="ntext" width="100%">Best&auml;m ett v&auml;rde p&aring; $\,x\,$ som g&ouml;r att uttrycket $\,4x^2-28x+48\,$ &auml;r negativt.</td>
</tr> </tr>
<tr> <tr>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">Ekvationen $\,x^2+4x+b=0\,$ har en rot $\,x=1\,$. Best&auml;m v&auml;rdet p&aring; konstanten $\,b\,$.</td> <td class="ntext" width="100%">Ekvationen $\,x^2+4x+b=0\,$ har en rot $\,x=1\,$. Best&auml;m v&auml;rdet p&aring; konstanten $\,b\,$.</td>
</tr> </tr>
Rad 107: Rad 107:
</div> </div>
-==&Ouml;vning 2.3:6==+'''&Ouml;vning 2.3:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m det minsta v&auml;rde som f&ouml;ljande polynom antar Best&auml;m det minsta v&auml;rde som f&ouml;ljande polynom antar
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$x^2-2x+1$</td> <td class="ntext" width="33%">$x^2-2x+1$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$x^2-4x+2$</td> <td class="ntext" width="33%">$x^2-4x+2$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$x^2-5x+7$</td> <td class="ntext" width="33%">$x^2-5x+7$</td>
</tr> </tr>
Rad 123: Rad 123:
</div> </div>
-==&Ouml;vning 2.3:7==+'''&Ouml;vning 2.3:7'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m det st&ouml;rsta v&auml;rde som f&ouml;ljande polynom antar Best&auml;m det st&ouml;rsta v&auml;rde som f&ouml;ljande polynom antar
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$1-x^2$</td> <td class="ntext" width="33%">$1-x^2$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$-x^2+3x-4$</td> <td class="ntext" width="33%">$-x^2+3x-4$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$x^2+x+1$</td> <td class="ntext" width="33%">$x^2+x+1$</td>
</tr> </tr>
Rad 139: Rad 139:
</div> </div>
-==&Ouml;vning 2.3:8==+'''&Ouml;vning 2.3:8'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Skissera grafen till f&ouml;ljande funktioner Skissera grafen till f&ouml;ljande funktioner
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$f(x)=x^2+1$</td> <td class="ntext" width="33%">$f(x)=x^2+1$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$f(x)=(x-1)^2+2$</td> <td class="ntext" width="33%">$f(x)=(x-1)^2+2$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$f(x)=x^2-6x+11$</td> <td class="ntext" width="33%">$f(x)=x^2-6x+11$</td>
</tr> </tr>
Rad 155: Rad 155:
</div> </div>
-==&Ouml;vning 2.3:9==+'''&Ouml;vning 2.3:9'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Hitta alla sk&auml;rningspunkter mellan x-axeln och kurvan Hitta alla sk&auml;rningspunkter mellan x-axeln och kurvan
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$y=x^2-1$</td> <td class="ntext" width="33%">$y=x^2-1$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$y=x^2-5x+6$</td> <td class="ntext" width="33%">$y=x^2-5x+6$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$y=3x^2-12x+9$</td> <td class="ntext" width="33%">$y=3x^2-12x+9$</td>
</tr> </tr>
Rad 171: Rad 171:
</div> </div>
-==&Ouml;vning 2.3:10==+'''&Ouml;vning 2.3:10'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Rita in i ett ''xy''-plan alla punkter vars koordinater $\,(x,y)\,$ uppfyller Rita in i ett ''xy''-plan alla punkter vars koordinater $\,(x,y)\,$ uppfyller
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$y \geq x^2\ $ och $\ y \leq 1 $</td> <td class="ntext" width="50%">$y \geq x^2\ $ och $\ y \leq 1 $</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$y \leq 1-x^2\ $ och $\ x \geq 2y-3 $</td> <td class="ntext" width="50%">$y \leq 1-x^2\ $ och $\ x \geq 2y-3 $</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$1 \geq x \geq y^2 $</td> <td class="ntext" width="50%">$1 \geq x \geq y^2 $</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$x^2 \leq y \leq x $</td> <td class="ntext" width="50%">$x^2 \leq y \leq x $</td>
</tr> </tr>
Rad 190: Rad 190:
</table> </table>
</div> </div>
 +<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>

Nuvarande version

Övning 2.3:1

Kvadratkomplettera följande uttryck

a)    $x^2-2x$ b)    $x^2+2x-1$ c)    $5+2x-x^2$ d)    $x^2+5x+3$

Övning 2.3:2

Lös följande andragradsekvationer med kvadratkomplettering

a)    $x^2-4x+3=0$ b)    $y^2+2y-15=0$ c)    $y^2+3y+4=0$
d)    $4x^2-28x+13=0$ e)    $5x^2+2x-3=0$ f)    $3x^2-10x+8=0$

Övning 2.3:3

Lös följande ekvationer direkt

a)    $x(x+3)=0$ b)    $(x-3)(x+5)=0$
c)    $5(3x-2)(x+8)=0$ d)    $x(x+3)-x(2x-9)=0$
e)    $(x+3)(x-1)-(x+3)(2x-9)=0$ f)    $x(x^2-2x)+x(2-x)=0$

Övning 2.3:4

Bestäm en andragradsekvation som har rötterna

a)    $-1\ $ och $\ 2$
b)    $1+\sqrt{3}\ $ och $\ 1-\sqrt{3}$
c)    $3\ $ och $\ \sqrt{3}$

Övning 2.3:5

a)    Bestäm en andragradsekvation som bara har $\,-7\,$ som rot.
b)    Bestäm ett värde på $\,x\,$ som gör att uttrycket $\,4x^2-28x+48\,$ är negativt.
c)    Ekvationen $\,x^2+4x+b=0\,$ har en rot $\,x=1\,$. Bestäm värdet på konstanten $\,b\,$.

Övning 2.3:6

Bestäm det minsta värde som följande polynom antar

a)    $x^2-2x+1$ b)    $x^2-4x+2$ c)    $x^2-5x+7$

Övning 2.3:7

Bestäm det största värde som följande polynom antar

a)    $1-x^2$ b)    $-x^2+3x-4$ c)    $x^2+x+1$

Övning 2.3:8

Skissera grafen till följande funktioner

a)    $f(x)=x^2+1$ b)    $f(x)=(x-1)^2+2$ c)    $f(x)=x^2-6x+11$

Övning 2.3:9

Hitta alla skärningspunkter mellan x-axeln och kurvan

a)    $y=x^2-1$ b)    $y=x^2-5x+6$ c)    $y=3x^2-12x+9$

Övning 2.3:10

Rita in i ett xy-plan alla punkter vars koordinater $\,(x,y)\,$ uppfyller

a)    $y \geq x^2\ $ och $\ y \leq 1 $ b)    $y \leq 1-x^2\ $ och $\ x \geq 2y-3 $
c)    $1 \geq x \geq y^2 $ d)    $x^2 \leq y \leq x $


































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