https://peda.net/id/0f629db2cd2 2018-10-11T10:27:08+03:00 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/04a499e8cd3 2018-10-11T11:38:24+03:00 <p>a) f, g, ne eivät leikkaa x-akselia<br/> b) f, g, niillä ei ole nollakohtia<br/> c) h, p, ne leikkaavat x-akselin<br/> d) h, sillä on yksi nollakohta<br/> e) h, sillä on yksi nollakohta</p> 2018-10-11T11:38:24+03:00 https://peda.net/id/da7e9d74ce6 2018-10-13T01:31:43+03:00 <div class="content enclose">a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D12" alt="f\left(1\right)=12"/></div> <div>selvitetään a</div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(x%2B2%5Cright)%5Cleft(x-5%5Cright)%3D12%7B%2C%7D%5C%20kun%5C%20x%3D1" alt="a\left(x+2\right)\left(x-5\right)=12{,}\ kun\ x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(1%2B2%5Cright)%5Cleft(1-5%5Cright)%3D12" alt="a\left(1+2\right)\left(1-5\right)=12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-12a%3D12%5C%20%5Cparallel%3A%5Cleft(-12%5Cright)" alt="-12a=12\ \parallel:\left(-12\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-1" alt="a=-1"/></div> <div>muutetaan funktioksi tekijöistä sieventämällä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cleft(x%2B2%5Cright)%5Cleft(x-5%5Cright)" alt="-1\left(x+2\right)\left(x-5\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cleft(x%5E2%2B2x-5x-10%5Cright)" alt="-1\left(x^2+2x-5x-10\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E2%2B3x%2B10" alt="-x^2+3x+10"/><br/> b) <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D12" alt="f\left(1\right)=12"/><br/> selvitetään tekijät nollakohdista<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-2%3D0" alt="x-2=0"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B1%7D%7B2%7D" alt="x=-\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-%5Cfrac%7B1%7D%7B2%7D%3D0" alt="x-\frac{1}{2}=0"/></div> <div>selvitetään a</div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(x-2%5Cright)%5Cleft(x%2B%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D12%7B%2C%7D%5C%20kun%5C%20x%3D1" alt="a\left(x-2\right)\left(x+\frac{1}{2}\right)=12{,}\ kun\ x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(1-2%5Cright)%5Cleft(1%2B%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D12" alt="a\left(1-2\right)\left(1+\frac{1}{2}\right)=12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5C%20%5Cfrac%7B1%7D%7B2%7Da%3D12%5C%20%5Cparallel%3A%5Cleft(-1%5C%20%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="-1\ \frac{1}{2}a=12\ \parallel:\left(-1\ \frac{1}{2}\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-8" alt="a=-8"/></div> <div>muodostetaan funktioksi tekijöistä sieventämällä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-8%5Cleft(x-2%5Cright)%5Cleft(x%2B%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="-8\left(x-2\right)\left(x+\frac{1}{2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-8%5Cleft(x%5E2%2B%5Cfrac%7B1%7D%7B2%7Dx-2x-1%5Cright)" alt="-8\left(x^2+\frac{1}{2}x-2x-1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-8x%5E2%2B12x%2B8" alt="-8x^2+12x+8"/><br/> c)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D12" alt="f\left(1\right)=12"/><br/> selvitetään tekijät nollakohdista<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3" alt="x=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-3%3D0" alt="x-3=0"/></div> <div>selvitetään a</div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(x-3%5Cright)%5Cleft(x-3%5Cright)%3D12%7B%2C%7D%5C%20kun%5C%20x%3D1" alt="a\left(x-3\right)\left(x-3\right)=12{,}\ kun\ x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(-2%5Cright)%5Cleft(-2%5Cright)%3D12" alt="a\left(-2\right)\left(-2\right)=12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4a%3D12%5C%20%5C%20%5Cparallel%3A4" alt="4a=12\ \ \parallel:4"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D3" alt="a=3"/></div> <div>muodostetaan funktioksi tekijöistä sieventämällä</div> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cleft(x-3%5Cright)%5Cleft(x-3%5Cright)" alt="3\left(x-3\right)\left(x-3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cleft(x%5E2-3x-3x%2B9%5Cright)" alt="3\left(x^2-3x-3x+9\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2-18x%2B27" alt="3x^2-18x+27"/></div> </div> </div> 2018-10-13T01:26:24+03:00 https://peda.net/id/d4c2950acd2 2018-10-11T11:29:55+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2-12x%2B12" alt="3x^2-12x+12"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B12%7D%7B6%7D%3D2" alt="x=\frac{12}{6}=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cleft(x-2%5Cright)%5E2" alt="3\left(x-2\right)^2"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%5E2%2B3x-4" alt="-2x^2+3x-4"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-3%5Cpm%5Csqrt%7B-23%7D%7D%7B-4%7D" alt="x=\frac{-3\pm\sqrt{-23}}{-4}"/></div> <div>yhtälöllä ei ole nollakohtia, se on jaoton</div> <div> </div> 2018-10-11T11:29:55+03:00 https://peda.net/id/72922f30cd2 2018-10-11T11:27:10+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%5E2-x-2" alt="6x^2-x-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%5Cpm7%7D%7B12%7D%3D%5Cfrac%7B2%7D%7B3%7D%5C%20tai%5C%20-%5Cfrac%7B1%7D%7B2%7D" alt="x=\frac{1\pm7}{12}=\frac{2}{3}\ tai\ -\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6%5Cleft(x-%5Cfrac%7B2%7D%7B3%7D%5Cright)%5Cleft(x%2B%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="6\left(x-\frac{2}{3}\right)\left(x+\frac{1}{2}\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=10x%5E2%2Bx-3" alt="10x^2+x-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%5Cpm11%7D%7B20%7D%3D%5Cfrac%7B1%7D%7B2%5C%20%7D%5C%20tai%5C%20-%5Cfrac%7B3%7D%7B5%7D" alt="x=\frac{-1\pm11}{20}=\frac{1}{2\ }\ tai\ -\frac{3}{5}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=10%5Cleft(x-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5Cleft(x%2B%5Cfrac%7B3%7D%7B5%7D%5Cright)" alt="10\left(x-\frac{1}{2}\right)\left(x+\frac{3}{5}\right)"/> 2018-10-11T11:27:10+03:00 https://peda.net/id/69c2dd24cd2 2018-10-11T11:19:46+03:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2%2B9x-5" alt="2x^2+9x-5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-9%5Cpm%5Csqrt%7B9%5E2-4%5Ccdot2%5Ccdot%5Cleft(-5%5Cright)%7D%7D%7B4%7D%3D%5Cfrac%7B-9%5Cpm%5Csqrt%7B121%7D%7D%7B4%7D" alt="x=\frac{-9\pm\sqrt{9^2-4\cdot2\cdot\left(-5\right)}}{4}=\frac{-9\pm\sqrt{121}}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B-9%2B11%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B2%7D%5C%20tai%5C%20%5Cfrac%7B-20%7D%7B4%7D%3D-5" alt="\frac{-9+11}{4}=\frac{1}{2}\ tai\ \frac{-20}{4}=-5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cleft(x-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5Cleft(x%2B5%5Cright)" alt="2\left(x-\frac{1}{2}\right)\left(x+5\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%5E2-7x-6" alt="3x^2-7x-6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B7%5Cpm%5Csqrt%7B121%7D%7D%7B6%7D%3D%5Cfrac%7B7%2B11%7D%7B6%7D%3D3%5C%20tai%5C%20%5Cfrac%7B7-11%7D%7B6%7D%3D-%5Cfrac%7B2%7D%7B3%7D" alt="x=\frac{7\pm\sqrt{121}}{6}=\frac{7+11}{6}=3\ tai\ \frac{7-11}{6}=-\frac{2}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Cleft(x-3%5Cright)%5Cleft(x%2B%5Cfrac%7B2%7D%7B3%7D%5Cright)" alt="3\left(x-3\right)\left(x+\frac{2}{3}\right)"/> 2018-10-11T11:19:46+03:00 https://peda.net/id/6dc743decd2 2018-10-11T11:12:43+03:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5Ccdot7%7D%7B2%7D%3D2%5Ccdot2%5Ccdot7%3D28" alt="\frac{2\cdot7}{2}=2\cdot2\cdot7=28"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(x-2%5Cright)%5Cleft(x%2B7%5Cright)%7D%7Bx-2%7D%3D%5Cleft(x-2%5Cright)%5E2%5Cleft(x%2B7%5Cright)%3D%5Cleft(x%5E2%2B4%5Cright)%5Cleft(x%2B7%5Cright)%3Dx%5E3%2B7x%5E2%2B4x%2B28" alt="\frac{\left(x-2\right)\left(x+7\right)}{x-2}=\left(x-2\right)^2\left(x+7\right)=\left(x^2+4\right)\left(x+7\right)=x^3+7x^2+4x+28"/><br/> c)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%5E2-x-12%7D%7Bx%2B3%7D" alt="\frac{x^2-x-12}{x+3}"/></div> <div>lasketaan nollakohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%7B49%7D%7D%7B2%7D%3D%5Cfrac%7B-1%2B49%7D%7B2%7D%3D24%5C%20tai%5C%20%5Cfrac%7B-1-49%7D%7B2%7D%3D-25" alt="x=\frac{-1\pm\sqrt{49}}{2}=\frac{-1+49}{2}=24\ tai\ \frac{-1-49}{2}=-25"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B24%5E2-24-12%7D%7B27%7D%3D%5Cfrac%7B540%7D%7B27%7D%3D20%5C%20" alt="\frac{24^2-24-12}{27}=\frac{540}{27}=20\ "/></div> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B-25%5E2-25-12%7D%7B-22%7D%3D%5Cfrac%7B588%7D%7B-22%7D%3D-26%7B%2C%7D72727...%5Capprox-27" alt="\frac{-25^2-25-12}{-22}=\frac{588}{-22}=-26{,}72727...\approx-27"/></div> </div> 2018-10-11T11:12:43+03:00 https://peda.net/id/45369a38cd2 2018-10-11T11:04:25+03:00 a)<br/> nollakohdat x=-2 ja x=1<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Da%5Cleft(x%2B2%5Cright)%5Cleft(x-1%5Cright)" alt="f\left(x\right)=a\left(x+2\right)\left(x-1\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(ax%2B2a%5Cright)%5Cleft(x-1%5Cright)%3D%5Cleft(ax%5E2%2Bax-2a%5Cright)" alt="\left(ax+2a\right)\left(x-1\right)=\left(ax^2+ax-2a\right)"/></div> <div>Funktion kuvaaja kulkee pisteen (0, 4) kautta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D4" alt="f\left(0\right)=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(0%2B2%5Cright)%5Cleft(0-1%5Cright)%3D4" alt="a\left(0+2\right)\left(0-1\right)=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2a%3D4" alt="-2a=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D-2" alt="a=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D-2x%5E2-2x%2B4" alt="f\left(x\right)=-2x^2-2x+4"/><br/> b)<br/> nollakohdat x=-1 ja x=3<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Da%5Cleft(x%2B1%5Cright)%5Cleft(x-3%5Cright)" alt="f\left(x\right)=a\left(x+1\right)\left(x-3\right)"/></div> <div>Funktion kuvaaja kulkee pisteen (0, -1) kautta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D-1" alt="f\left(0\right)=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(0%2B1%5Cright)%5Cleft(0-3%5Cright)%3D-1" alt="a\left(0+1\right)\left(0-3\right)=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-3a%3D-1" alt="-3a=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7B1%7D%7B3%7D" alt="a=\frac{1}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B3%7D%5Cleft(x%2B1%5Cright)%5Cleft(x-3%5Cright)" alt="\frac{1}{3}\left(x+1\right)\left(x-3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B1%7D%7B3%7Dx%2B%5Cfrac%7B1%7D%7B3%7D%5Cright)%5Cleft(x-3%5Cright)%3D%5Cfrac%7B1%7D%7B3%7Dx%5E2-x%2B%5Cfrac%7B1%7D%7B3%7Dx-1" alt="\left(\frac{1}{3}x+\frac{1}{3}\right)\left(x-3\right)=\frac{1}{3}x^2-x+\frac{1}{3}x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B3%7Dx%5E2-%5Cfrac%7B2%7D%7B3%7Dx-1" alt="\frac{1}{3}x^2-\frac{2}{3}x-1"/></div> </div> 2018-10-11T11:04:25+03:00 https://peda.net/id/e04b7682cd2 2018-10-11T10:40:12+03:00 <a href="https://peda.net/p/oskari.lahtinen/mpjy/3-4/382/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mpjy/3-4/382/sieppaa-png:file/photo/802bb443a83bb7bf2031ee6c0e25dc4a2a7c8e98/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a> 2018-10-11T10:40:08+03:00 https://peda.net/id/ac448c94cd2 2018-10-11T10:31:31+03:00 <div>Jaa tekijöihin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2-3x%2B1" alt="2x^2-3x+1"/></div> Ratkaistaan nollakohdat<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" alt="x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%5Cpm%5Csqrt%7B1%7D%7D%7B4%7D%3D%5Cfrac%7B3%2B1%7D%7B4%7D%3D1%5C%20tai%5C%20%5Cfrac%7B3-1%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\frac{3\pm\sqrt{1}}{4}=\frac{3+1}{4}=1\ tai\ \frac{3-1}{4}=\frac{1}{2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Cleft(x-x_1%5Cright)%5Cleft(x-x_2%5Cright)" alt="a\left(x-x_1\right)\left(x-x_2\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cleft(x-1%5Cright)%5Cleft(x-%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="2\left(x-1\right)\left(x-\frac{1}{2}\right)"/></div> <div>tarkistus:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cleft(x-1%5Cright)%5Cleft(x-%5Cfrac%7B1%7D%7B2%7D%5Cright)%3D%5Cleft(2x-2%5Cright)%5Cleft(x-%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="2\left(x-1\right)\left(x-\frac{1}{2}\right)=\left(2x-2\right)\left(x-\frac{1}{2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E2-2x-x%2B1%3D2x%5E2-3x%2B1" alt="2x^2-2x-x+1=2x^2-3x+1"/></div> 2018-10-11T10:31:31+03:00