Finding the graph of a function based on its propertiesI know the following things about my function: f ( – 4 ) = f ( 2 ) = 0 ; f ′ ( − 1 ) = 0 , f ′ ( x ) &lt; 0 for x &lt; − 1 ; f ′ ( x ) &gt; 0 for x &gt; − 1That is, it appears to be a quadratic function with intercepts at 2 and -4.How can I determine its equation?

Question

Finding the graph of a function based on its propertiesI know the following things about my function:   f  (  –  4  )  =  f  (  2  )  =  0  ;      f    ′    (  −  1  )  =  0  ,      f    ′    (  x  )  &amp;lt;  0   for   x  &amp;lt;  −  1  ;      f    ′    (  x  )  &amp;gt;  0   for   x  &amp;gt;  −  1That is, it appears to be a quadratic function with intercepts at 2 and -4.How can I determine its equation?

Ismael Wilkinson · Accepted Answer

Step 1Consider any function of the form       f    n    (  x  )  =  (  x  +  1      )          2      n        −      3          2      n      If we substitute any natural number for n in the above equation, the resulting equation will fulfill all of the descriptions you gave. So, your description describes a set of functions, not a single one.Step 2  n  =  1 would yield the quadratic you&#039;re looking for:       f    1    (  x  )  =  (  x  +  1      )    2    −  9  =  (  x  +  4  )  (  x  −  2  ). The derivative of this is       f    1    ′    (  x  )  =  2  (  x  +  1  ), which will be 0 at   x  =  −  1, negative whenever   x  &amp;lt;  −  1 and positive when   x  &amp;gt;  −  1.

Finding the graph of a function based on its properties. I know the following things about my function: f(-4)=f(2)=0; f′(-1)=0, f′(x)<0 for x<-1; f′(x)>0 for x>-1

Answered question

Answer & Explanation

New Questions in Algebra I