https://peda.net/id/ca46e3f6db6 2018-10-29T12:45:45+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/0dddb6cadb7 2018-10-29T14:06:23+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%2B6%3D0" alt="3x+6=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D-6%5C%20%5Cparallel%3A3" alt="3x=-6\ \parallel:3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-2" alt="x=-2"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4x-5%3D2x%2B6" alt="4x-5=2x+6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4x-2x%3D6%2B5" alt="4x-2x=6+5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D11%5C%20%5Cparallel%3A2" alt="2x=11\ \parallel:2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D5%7B%2C%7D5" alt="x=5{,}5"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x-2%5Cright)%5Cleft(2x-1%5Cright)%3D0" alt="\left(3x-2\right)\left(2x-1\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%5E2-7x%2B2%3D0" alt="6x^2-7x+2=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-b%5Cpm%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%3D%5Cfrac%7B7%5Cpm%5Csqrt%7B%5Cleft(-7%5Cright)%5E2-4%5Ccdot6%5Ccdot2%7D%7D%7B12%7D%3D%5Cfrac%7B7%5Cpm%5Csqrt%7B49-48%7D%7D%7B12%7D%3D%5Cfrac%7B7%5Cpm1%7D%7B12%7D%3D%5Cfrac%7B6%7D%7B12%7D%3D%5Cfrac%7B1%7D%7B2%7D%5C%20tai%5C%20%5Cfrac%7B8%7D%7B12%7D%3D%5Cfrac%7B2%7D%7B3%7D" alt="x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{7\pm\sqrt{\left(-7\right)^2-4\cdot6\cdot2}}{12}=\frac{7\pm\sqrt{49-48}}{12}=\frac{7\pm1}{12}=\frac{6}{12}=\frac{1}{2}\ tai\ \frac{8}{12}=\frac{2}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x-2%5Cright)%5Cleft(2x-1%5Cright)%3D2" alt="\left(3x-2\right)\left(2x-1\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%5E2-7x%3D0" alt="6x^2-7x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cleft(6x-7%5Cright)%3D0" alt="x\left(6x-7\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%5C%20tai%5C%206x-7%3D0" alt="x=0\ tai\ 6x-7=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%3D7%5C%20%5Cparallel%3A6" alt="6x=7\ \parallel:6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%5C%20%5Cfrac%7B1%7D%7B7%7D" alt="x=1\ \frac{1}{7}"/> 2018-10-29T14:06:23+02:00 https://peda.net/id/2e9841c0db7 2018-10-29T13:52:59+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E5%2B12x%5E3%3D3x%5E3%5Cleft(x%5E2%2B4%5Cright)" alt="3x^5+12x^3=3x^3\left(x^2+4\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E5%2B12x%5E3%3D3x%5E3%5Cleft(x%5E2-4%5Cright)" alt="3x^5+12x^3=3x^3\left(x^2-4\right)"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3-2x%5E2%2B3x-6%3Dx%5E2%5Cleft(x-2%5Cright)%2B3%5Cleft(x-2%5Cright)%3D%5Cleft(x-2%5Cright)%5Cleft(x%5E2%2B3%5Cright)" alt="x^3-2x^2+3x-6=x^2\left(x-2\right)+3\left(x-2\right)=\left(x-2\right)\left(x^2+3\right)"/><br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4-1%3D%5Cleft(x%5E2%2B1%5Cright)%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)" alt="x^4-1=\left(x^2+1\right)\left(x-1\right)\left(x+1\right)"/> 2018-10-29T13:52:59+02:00 https://peda.net/id/048b5b2adb7 2018-10-29T13:44:39+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B6x%3D2x%5Cleft(x%2B3%5Cright)" alt="2x^2+6x=2x\left(x+3\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-4%3D%5Cleft(x%2B2%5Cright)%5Cleft(x-2%5Cright)" alt="x^2-4=\left(x+2\right)\left(x-2\right)"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-2x%2B1%3D%5Cleft(x-1%5Cright)%5E2" alt="x^2-2x+1=\left(x-1\right)^2"/><br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=9x%5E2-25%3D%5Cleft(3x-5%5Cright)%5Cleft(3x-5%5Cright)" alt="9x^2-25=\left(3x-5\right)\left(3x-5\right)"/> 2018-10-29T13:44:39+02:00 https://peda.net/id/6e4fa658db6 2018-10-29T13:33:17+02:00 a) nollakohtia voi olla korkeintaan asteluvun verran<br/> b) onko suora nouseva vai laskeva<br/> c) onko paraabeli ylöspäin vai alaspäin avautuva<br/> d) missä paraabelin huippu on 2018-10-29T13:33:17+02:00 https://peda.net/id/ba204a8adb6 2018-10-29T13:21:06+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2-2x-3" alt="f\left(x\right)=x^2-2x-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-b%5Cpm%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B%5Cleft(-2%5Cright)%5E2-4%5Ccdot1%5Ccdot%5Cleft(-3%5Cright)%7D%7D%7B2%7D%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B16%7D%7D%7B2%7D%3D%5Cfrac%7B2%5Cpm4%7D%7B2%7D%3D%5Cfrac%7B2-4%7D%7B2%7D%3D-1%5C%20tai%5C%20%5Cfrac%7B2%2B4%7D%7B2%7D%3D%5Cfrac%7B6%7D%7B2%7D%3D3" alt="x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{2\pm\sqrt{\left(-2\right)^2-4\cdot1\cdot\left(-3\right)}}{2}=\frac{2\pm\sqrt{16}}{2}=\frac{2\pm4}{2}=\frac{2-4}{2}=-1\ tai\ \frac{2+4}{2}=\frac{6}{2}=3"/><br/> <div>väite on tosi, funktion nollakohdat ovat -1 ja 3</div> <div>b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%5E2-2" alt="f\left(x\right)=2x^2-2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-2%5Cright)%3D2%5Cleft(-2%5Cright)%5E2-2%3D2%5Cleft(4%5Cright)-2%3D8-2%3D6" alt="f\left(-2\right)=2\left(-2\right)^2-2=2\left(4\right)-2=8-2=6"/></div> <div>väite on epätosi, funktion arvo kohdassa x=-2 ei ole -10</div> <div>c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D7x-x%5E3" alt="f\left(x\right)=7x-x^3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(-1%5Cright)%3D7%5Cleft(-1%5Cright)-%5Cleft(-1%5Cright)%5E3%3D-7%2B1%3D-6" alt="f\left(-1\right)=7\left(-1\right)-\left(-1\right)^3=-7+1=-6"/></div> <div>väite on epätosi, koska funktion kohdassa x=-1, se ei saa arvoa -8</div> <div>d)</div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E4%2B1" alt="f\left(x\right)=x^4+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4%2B1%3D0" alt="x^4+1=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E4%3D-1%5C%20%5Csqrt%5B4%5D%7B%7D" alt="x^4=-1\ \sqrt[4]{}"/></div> <div>väite on tosi, negatiivisesta luvusta ei voi ottaa parillista juurta eli yhtälöllä ei ole nollakohtia</div> </div> 2018-10-29T13:21:06+02:00 https://peda.net/id/7462fb8edb6 2018-10-29T13:04:50+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x%5Cleft(3x%2B4%5Cright)%2B1%3D6x%5E2%2B8x%2B1" alt="f\left(x\right)=2x\left(3x+4\right)+1=6x^2+8x+1"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3%5Cleft(2x-1%5Cright)%2Bx%5Cleft(-4x%2B5%5Cright)%3D6x-3%2B%5Cleft(-4x%5E2%5Cright)%2B5x%3D-4x%5E2%2B11x-3" alt="f\left(x\right)=3\left(2x-1\right)+x\left(-4x+5\right)=6x-3+\left(-4x^2\right)+5x=-4x^2+11x-3"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-%5Cleft(2x-1%5Cright)%5Cleft(x%2B3%5Cright)%3D-%5Cleft(2x%5E2%2B6x-x-3%5Cright)%3D-2x%5E2-5x%2B3" alt="f\left(x\right)=-\left(2x-1\right)\left(x+3\right)=-\left(2x^2+6x-x-3\right)=-2x^2-5x+3"/><br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D1-%5Cleft(3x%5E4%2B1%5Cright)%5E2%3D1-3x%5E8-1%3D3x%5E8" alt="f\left(x\right)=1-\left(3x^4+1\right)^2=1-3x^8-1=3x^8"/> 2018-10-29T13:04:50+02:00 https://peda.net/id/24f50516db6 2018-10-29T12:55:27+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2x%2B3%5Cright)%5E2%2B3x%5E2%3D2x%5E2%2B3%5E2%2B3x%5E2%3D5x%5E2%2B9" alt="\left(2x+3\right)^2+3x^2=2x^2+3^2+3x^2=5x^2+9"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3-x%5Cright)%5Cleft(3%2Bx%5Cright)-x%5E2%3D%5Cleft(3%5E2-x%5E2%5Cright)-x%5E2%3D-2x%5E2%2B9" alt="\left(3-x\right)\left(3+x\right)-x^2=\left(3^2-x^2\right)-x^2=-2x^2+9"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%5E3%2B1%5Cright)%5Cleft(x%5E3-1%5Cright)%3D%5Cleft(x%5E3%5Cright)%5E2-1%5E2%3Dx%5E6-1" alt="\left(x^3+1\right)\left(x^3-1\right)=\left(x^3\right)^2-1^2=x^6-1"/><br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%5Cleft(x%5E2%2B1%5Cright)%3D%5Cleft(x%5E2-1%5E2%5Cright)%5Cleft(x%5E2%2B1%5Cright)%3D%5C%20x%5E4%2Bx%5E2-x%5E2-1%3Dx%5E4-1" alt="\left(x+1\right)\left(x-1\right)\left(x^2+1\right)=\left(x^2-1^2\right)\left(x^2+1\right)=\ x^4+x^2-x^2-1=x^4-1"/> 2018-10-29T12:55:27+02:00 https://peda.net/id/3588550adb6 2018-10-29T12:48:45+02:00 a) E<br/> b) A<br/> c) D<br/> d) B<br/> e) F<br/> f) C 2018-10-29T12:48:45+02:00