Question

1. on the following set of axes, create the graph of a function ( f(x) ) with the following characteristics:

( f(-7)=-1 ) and ( f(9)=8 )
zeros at: ( x=-6,1 ), and 7
( y - ) intercept ( =2 )
turning points at: ( (-3,7) ) and ( (4,-4) )

1. explain why this function must be decreasing from ( x=-3 ) to ( x = 4 ).

Explanation:

Step1: Recall the definition of a decreasing function

A function \(y = f(x)\) is decreasing on an interval \((a,b)\) if for any two points \(x_1,x_2\in(a,b)\) with \(x_1

Step2: Analyze the turning points

We are given two turning points \((-3,7)\) and \((4, - 4)\). At a local maximum (in this case, the point \((-3,7)\)), the function changes from increasing to decreasing. At a local minimum (in this case, the point \((4,-4)\)), the function changes from decreasing to increasing.
Since \((-3,7)\) is a local maximum (the \(y\) - value is higher than the neighboring points in its immediate vicinity) and \((4,-4)\) is a local minimum (the \(y\) - value is lower than the neighboring points in its immediate vicinity), for all \(x\) values such that \(-3

Answer:

The function has a local maximum at \(x=-3\) (where the function changes from increasing to decreasing) and a local minimum at \(x = 4\) (where the function changes from decreasing to increasing). By the definition of increasing and decreasing functions (a function is decreasing on an interval if for \(x_1