SOLUTION: Write an equation for a quadratic function whose graph has x-intercepts of 3 and 7, and f(5)=8

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Write an equation for a quadratic function whose graph has x-intercepts of 3 and 7, and f(5)=8      Log On


   

Question 120926: Write an equation for a quadratic function whose graph has x-intercepts of 3 and 7, and f(5)=8
Answer by nabla(475) About Me  (Show Source):
You can put this solution on YOUR website!
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.
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.
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:
g%28x%29=-2%28x%5E2-10x%2B21%29
Check:
g%285%29=-2%2825-50%2B21%29=-2%2A-4=8
Note that g%28x%29=-2+f%28x%29, AND the zeros remain unchanged. For a simple graphical example, look at the following:
graph%28300%2C200%2C-1%2C10%2C-5%2C6%2Cx%5E2-10x%2B21%2C-2%28x%5E2-10x%2B21%29%29