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?