*Mathematics Teacher.*

The first think I see when I start this problem is the the function f(x) has three unknowns, x, b, and c. So right away I know before I can find f(4) I first need to determine what the values of b and c are. I also see that I am given two other pieces of information that might help in figuring out the values of b and c in this situation. The first thing I am going to do is use the fact that f(1)=9 and I am going to substitute 1 in place of x in my function f(x).

So from this information I have found out that when x=1 then f(x)=1+b+c. I also know that f(1)=9 because I was given that information at the beginning of the problem.

Now I see that I still have two unknowns b and c, so I still need to use the other information I have to determine the values of these unknowns.

I was given at the beginning of the problem that f(3)-f(2)=8. Even though I do not know exactly what f(3) or f(2) equal I do know that their difference is equal to 8. So now in the same way that I found f(1) earlier I am going to find f(2) and f(3).

So now I know what f(3) and f(2) are equal to in terms of b and c, and therefore I can find f(3)-f(2).

So now I have found that f(3)-f(2)=b+5. I remember that I was also told that f(3)-f(2)=8.

I have a value for b, this is progress! Now that I know that b=3, I can use this information and substitute b into my earlier equation, b+c=8. (Remember that I found b+c=8 when I found out that f(1)=9)

So now I know that b=3 and c=5. Since I know these values I can "complete" my function f(x) by subtituting in these values for b and c.

From here I can easily figure out the value for f(4) by substituing 4 in for x in my function f(x).

So I have finally finished my problem and found out that f(4)=33 when f(1)=9, and f(3)-f(2)=8.

If I were doing this problem in my classroom here are some follow up questions I might ask:

*1. Do the values of b and c change if we were given that f(1)=3? What if f(1)=-9?*

*2. How does the value for f(4) change if we were given that f(2)-f(3)=8?*

*3. Graph the functions f(x)=x^2+3x+5 and f(x)=x^2-3x-5 on the same graph. What happens to our original graph when b and c are both negative?*

Nice thinking through a problem. I don't think you have to step by step through the algebra so much - since that's not where the thinking really is. Would would really complete this post is to consider alternate strategies at the beginning or end, or to use this to pose another problem.

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