The given areTwo x-intercepts y-intercept math ( 0 , − 4 )Maximum at ( 2 , 4 )How to create quadratic equation given y intercept, and maximum and B = 8?

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The given areTwo   x-intercepts  y-intercept math   (  0  ,  −  4  )Maximum at   (  2  ,  4  )How to create quadratic equation given   y  intercept, and maximum and    B  =  8?

embraci4i · Accepted Answer

You can start by writing out the general form of a quadratic polynomial:   P  (  x  )  =  a      x    2    +  b  x  +  c. Saying that the y-intercept is   (  0  ,  −  4  ) is the same as saying that   P  (  0  )  =  −  4, which gives us   c  =  −  4. Now the derivative of the polynomial will be       P    ′    (  x  )  =  2  a  x  +  b. Since the derivative of a differentiable function at a maximum is   0, you must have       P    ′    (  2  )  =  0 or   4  a  =  −  b. Since the point   (  2  ,  4  ) is assumed to be in the graph of the function, we know that   P  (  2  )  =  4 or   4  a  +  2  b  −  4  =  4, which means that   4  a  =  8  −  2  b. Thus we have   4  a  =  −  b          and           4  a  =  8  −  2  bWhich gives   b  =  8 and   a  =  −  2, so we get   P  (  x  )  =  −  2      x    2    +  8  x  −  4

deiluefniwf · Accepted Answer

Another way to do it, would be to use the fact that every quadratic equation is of the form   P  (  x  )  =  α  (  x  −  β      )    2    +  γ for some   α  ,    β  ,    γ. When the equation is written in this form it becomes clear that   α must be negative (otherwise the function won&#039;t have a maximum), and then the maximum is attained when   x  −  β  =  0. Since the maximum is at   x  =  2, we have   β  =  2. Also, the value of   P at the maximum will be   4  =  P  (  2  )  =  α  (  0      )    2    +  γ  =  γ, so   γ  =  4. Lastly, we know that the y-intercept is   (  0  ,  −  4  ), so that   −  4  =  P  (  0  )  =  α  (  0  −  β      )    2    +  γ  =  4  α  +  4, which gives   α  =  −  2. Now we can expand the original expression and we get   P  (  x  )  =  α  (  x  −  β      )    2    +  γ  =  −  2  (  x  −  2      )    2    +  4  =  −  2      x    2    +  8  x  −  4, so we get the same result as before.

The given are Two x-intercepts y-intercept math (0,−4) Maximum at (2,4) How to create quadratic equation given y intercept, and maximum and B=8?

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