document.write( "Question 120926: Write an equation for a quadratic function whose graph has x-intercepts of 3 and 7, and f(5)=8 \n" ); document.write( "

Algebra.Com's Answer #88738 by nabla(475) $\"\"$ $\"About$

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First, notice that x-intercepts in a quadratic function are easily identified when we set the function =0. After factoring such a function, we will get something resembling: $\"%28x-a%29%28x-b%29=0\"$ . That equation, to be true, must have either factor =0. That is, $\"x-a=0\"$ or $\"x-b=0\"$ . Then, the x-intercepts are a and b because they satisfy each equation.\r
\n" ); document.write( "\n" ); document.write( "You should now be able to see that we are looking for a modified version of the following equation: $\"%28x-3%29%28x-7%29=0\"$ . When we expand this, we get $\"x%5E2-10x%2B21=0\"$ . Let the left side be our $\"f%28x%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "Then, $\"f%28x%29=x%5E2-10x%2B21\"$ . If we take $\"f%285%29=5%5E2-10%285%29%2B21=25-50%2B24=-4\"$ . Then, what should we multiply -4 by to get an answer of 8? $\"-4a=8\"$ implies $\"a=-2\"$ . Now, multiply the right side of the $\"f%28x%29\"$ we found by this $\"a=-2\"$ , to form a new function $\"g%28x%29\"$ :\r
\n" ); document.write( "\n" ); document.write( " $\"g%28x%29=-2%28x%5E2-10x%2B21%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Check:
\n" ); document.write( " $\"g%285%29=-2%2825-50%2B21%29=-2%2A-4=8\"$ \r
\n" ); document.write( "\n" ); document.write( "Note that $\"g%28x%29=-2+f%28x%29\"$ , AND the zeros remain unchanged. For a simple graphical example, look at the following:\r
\n" ); document.write( "\n" ); document.write( " $\"graph%28300%2C200%2C-1%2C10%2C-5%2C6%2Cx%5E2-10x%2B21%2C-2%28x%5E2-10x%2B21%29%29\"$ \n" ); document.write( "

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